Multi-axis interferometer with procedure and data processing for mirror mapping

ABSTRACT

In general, in one aspect, the invention features methods that include locating a plurality of alignment marks on a moveable stage, interferometrically measuring a position of a measurement object along an interferometer axis for each of the alignment mark locations, and using the interferometric position measurements to derive information about a surface figure of the measurement object. The position of the measurement object is measured using an interferometry assembly and either the measurement object or the interferometry assembly are attached to the stage.

CROSS-REFERENCE TO RELATED APPLICATIONS

Under 35 U.S.C. § 119(e)(1), this application claims priority toProvisional Patent Application 60/663,408, entitled “MULTI-AXISINTERFEROMETER WITH PROCEDURE AND DATA PROCESSING FOR MIRROR MAPPING,”filed Mar. 18, 2005, the entire contents of which are herebyincorporated by reference.

TECHNICAL FIELD

This disclosure relates to interferometers, and systems and methods thatuse interferometers.

BACKGROUND

Displacement measuring interferometers monitor changes in the positionof a measurement object relative to a reference object based on anoptical interference signal. The interferometer generates the opticalinterference signal by overlapping and interfering a measurement beamreflected from the measurement object with a reference beam reflectedfrom the reference object.

In many applications, the measurement and reference beams haveorthogonal polarizations and different frequencies. The differentfrequencies can be produced, for example, by laser Zeeman splitting, byacousto-optical modulation, or internal to the laser using birefringentelements or the like. The orthogonal polarizations allow a polarizingbeam-splitter to direct the measurement and reference beams to themeasurement and reference objects, respectively, and combine thereflected measurement and reference beams to form overlapping exitmeasurement and reference beams. The overlapping exit beams form anoutput beam that subsequently passes through a polarizer.

The polarizer mixes polarizations of the exit measurement and referencebeams to form a mixed beam. Components of the exit measurement andreference beams in the mixed beam interfere with one another so that theintensity of the mixed beam varies with the relative phase of the exitmeasurement and reference beams.

A detector measures the time-dependent intensity of the mixed beam andgenerates an electrical interference signal proportional to thatintensity. Because the measurement and reference beams have differentfrequencies, the electrical interference signal includes a “heterodyne”signal having a beat frequency equal to the difference between thefrequencies of the exit measurement and reference beams. If the lengthsof the measurement and reference paths are changing relative to oneanother, e.g., by translating a stage that includes the measurementobject, the measured beat frequency includes a Doppler shift equal to2vnp/λ, where v is the relative speed of the measurement and referenceobjects, λ is the wavelength of the measurement and reference beams, nis the refractive index of the medium through which the light beamstravel, e.g., air or vacuum, and p is the number of passes to thereference and measurement objects. Changes in the phase of the measuredinterference signal correspond to changes in the relative position ofthe measurement object, e.g., a change in phase of 2π correspondssubstantially to a distance change d of λ/(2np). Distance 2d is around-trip distance change or the change in distance to and from a stagethat includes the measurement object. In other words, the phase Φ,ideally, is directly proportional to d, and can be expressed as

$\begin{matrix}{{\Phi = {2\;{pkd}}},{{{where}\mspace{14mu} k} = {\frac{2\;\pi\; n}{\lambda}.}}} & (1)\end{matrix}$

Unfortunately, the observable interference phase, {tilde over (Φ)}, isnot always identically equal to phase Φ. Many interferometers include,for example, non-linearities such as “cyclic errors.” Cyclic errors canbe expressed as contributions to the observable phase and/or theintensity of the measured interference signal and have a sinusoidaldependence on the change in for example optical path length 2pnd. Inparticular, a first order cyclic error in phase has for the example asinusoidal dependence on (4πpnd)/λ and a second order cyclic error inphase has for the example a sinusoidal dependence on 2(4πpnd)/λ. Higherorder cyclic errors can also be present as well as sub-harmonic cyclicerrors and cyclic errors that have a sinusoidal dependence of otherphase parameters of an interferometer system comprising detectors andsignal processing electronics. Different techniques for quantifying suchcyclic errors are described in commonly owned U.S. Pat. No. 6,137,574,U.S. Pat. No. 6,252,688, and U.S. Pat. No. 6,246,481 by Henry A. Hill.

In addition to cyclic errors, there are other sources of deviations inthe observable interference phase from Φ, such as, for example,non-cyclic non-linearities or non-cyclic errors. One example of a sourceof a non-cyclic error is the diffraction of optical beams in themeasurement paths of an interferometer. Non-cyclic error due todiffraction has been determined for example by analysis of the behaviorof a system such as found in the work of J.-P. Monchalin, M. J. Kelly,J. E. Thomas, N. A. Kurnit, A. Szöke, F. Zernike, P. H. Lee, and A.Javan, “Accurate Laser Wavelength Measurement With A Precision Two-BeamScanning Michelson Interferometer,” Applied Optics, 20(5), 736-757,1981.

A second source of non-cyclic errors is the effect of “beam shearing” ofoptical beams across interferometer elements and the lateral shearing ofreference and measurement beams one with respect to the other. Beamshears can be caused, for example, by a change in direction ofpropagation of the input beam to an interferometer or a change inorientation of the object mirror in a double pass plane mirrorinterferometer such as a differential plane mirror interferometer (DPMI)or a high stability plane mirror interferometer (HSPMI).

Inhomogeneities in the interferometer optics may cause wavefront errorsin the reference and measurement beams. When the reference andmeasurement beams propagate collinearly with one another through suchinhomogeneities, the resulting wavefront errors are identical and theircontributions to the interferometric signal cancel each other out. Moretypically, however, the reference and measurement beam components of theoutput beam are laterally displaced from one another, i.e., they have arelative beam shear. Such beam shear causes the wavefront errors tocontribute an error to the interferometric signal derived from theoutput beam.

Moreover, in many interferometry systems beam shear changes as theposition or angular orientation of the measurement object changes. Forexample, a change in relative beam shear can be introduced by a changein the angular orientation of a plane mirror measurement object.Accordingly, a change in the angular orientation of the measurementobject produces a corresponding error in the interferometric signal.

The effect of the beam shear and wavefront errors will depend uponprocedures used to mix components of the output beam with respect tocomponent polarization states and to detect the mixed output beam togenerate an electrical interference signal. The mixed output beam mayfor example be detected by a detector without any focusing of the mixedbeam onto the detector, by detecting the mixed output beam as a beamfocused onto a detector, or by launching the mixed output beam into asingle mode or multi-mode optical fiber and detecting a portion of themixed output beam that is transmitted by the optical fiber. The effectof the beam shear and wavefront errors will also depend on properties ofa beam stop should a beam stop be used in the procedure to detect themixed output beam. Generally, the errors in the interferometric signalare compounded when an optical fiber is used to transmit the mixedoutput beam to the detector.

Amplitude variability of the measured interference signal can be the netresult of a number of mechanisms. One mechanism is a relative beam shearof the reference and measurement components of the output beam that isfor example a consequence of a change in orientation of the measurementobject.

In dispersion measuring applications, optical path length measurementsare made at multiple wavelengths, e.g., 532 nm and 1064 nm, and are usedto measure dispersion of a gas in the measurement path of the distancemeasuring interferometer. The dispersion measurement can be used inconverting the optical path length measured by a distance measuringinterferometer into a physical length. Such a conversion can beimportant since changes in the measured optical path length can becaused by gas turbulence and/or by a change in the average density ofthe gas in the measurement arm even though the physical distance to themeasurement object is unchanged.

The interferometers described above are often components of metrologysystems in scanners and steppers used in lithography to produceintegrated circuits on semiconductor wafers. Such lithography systemstypically include a translatable stage to support and fix the wafer,focusing optics used to direct a radiation beam onto the wafer, ascanner or stepper system for translating the stage relative to theexposure beam, and one or more interferometers. Each interferometerdirects a measurement beam to, and receives a reflected measurement beamfrom, e.g., a plane mirror attached to the stage. Each interferometerinterferes its reflected measurement beams with a correspondingreference beam, and collectively the interferometers accurately measurechanges in the position of the stage relative to the radiation beam. Theinterferometers enable the lithography system to precisely control whichregions of the wafer are exposed to the radiation beam.

In many lithography systems and other applications, the measurementobject includes one or more plane mirrors to reflect the measurementbeam from each interferometer. Small changes in the angular orientationof the measurement object, e.g., corresponding to changes in thepitching and/or yaw of a stage, can alter the direction of eachmeasurement beam reflected from the plane mirrors. If leftuncompensated, the altered measurement beams reduce the overlap of theexit measurement and reference beams in each correspondinginterferometer. Furthermore, these exit measurement and reference beamswill not be propagating parallel to one another nor will their wavefronts be aligned when forming the mixed beam. As a result, theinterference between the exit measurement and reference beams will varyacross the transverse profile of the mixed beam, thereby corrupting theinterference information encoded in the optical intensity measured bythe detector.

To address this problem, many conventional interferometers include aretroreflector that redirects the measurement beam back to the planemirror so that the measurement beam “double passes” the path between theinterferometer and the measurement object. The presence of theretroreflector insures that the direction of the exit measurement isinsensitive to changes in the angular orientation of the measurementobject. When implemented in a plane mirror interferometer, theconfiguration results in what is commonly referred to as ahigh-stability plane mirror interferometer (HSPMI). However, even withthe retroreflector, the lateral position of the exit measurement beamremains sensitive to changes in the angular orientation of themeasurement object. Furthermore, the path of the measurement beamthrough optics within the interferometer also remains sensitive tochanges in the angular orientation of the measurement object.

In practice, the interferometry systems are used to measure the positionof the wafer stage along multiple measurement axes. For example,defining a Cartesian coordinate system in which the wafer stage lies inthe x-y plane, measurements are typically made of the x and y positionsof the stage as well as the angular orientation of the stage withrespect to the z axis, as the wafer stage is translated along the x-yplane. Furthermore, it may be desirable to also monitor tilts of thewafer stage out of the x-y plane. For example, accurate characterizationof such tilts may be necessary to calculate Abbé offset errors in the xand y positions. Thus, depending on the desired application, there maybe up to five degrees of freedom to be measured. Moreover, in someapplications, it is desirable to also monitor the position of the stagewith respect to the z-axis, resulting in a sixth degree of freedom.

To measure each degree of freedom, an interferometer is used to monitordistance changes along a corresponding metrology axis. For example, insystems that measure the x and y positions of the stage as well as theangular orientation of the stage with respect to the x, y, and z axes,at least three spatially separated measurement beams reflect from oneside of the wafer stage and at least two spatially separated measurementbeams reflect from another side of the wafer stage. See, e.g., U.S. Pat.No. 5,801,832 entitled “METHOD OF AND DEVICE FOR REPETITIVELY IMAGING AMASK PATTERN ON A SUBSTRATE USING FIVE MEASURING AXES,” the contents ofwhich are incorporated herein by reference. Each measurement beam isrecombined with a reference beam to monitor optical path length changesalong the corresponding metrology axes. Because the differentmeasurement beams contact the wafer stage at different locations, theangular orientation of the wafer stage can then be derived fromappropriate combinations of the optical path length measurements.Accordingly, for each degree of freedom to be monitored, the systemincludes at least one measurement beam that contacts the wafer stage.Furthermore, as described above, each measurement beam may double-passthe wafer stage to prevent changes in the angular orientation of thewafer stage from corrupting the interferometric signal. The measurementbeams may be generated from physically separate interferometers or frommulti-axes interferometers that generate multiple measurement beams.

SUMMARY

In addition to the sources of interferometer measurement errorsdiscussed above, another source of measurement error in certaindisplacement measuring interferometry systems are variations in thesurface of a plane mirror measurement object. These variations cause themirror surface to deviate from being perfectly planar. The mirrorsurface shape is referred to as its surface figure, which can becharacterized by a mirror surface figure function. Errors ininterferometry systems that use plane mirror measurement objects can bereduced where information about the surface figure of the mirror isknown. For example, knowledge of the surface figure functions allows thesystem to compensate for the variations in the surface figure from anideal mirror. However, the surface figure of plane mirrors can vary withtime, so the accuracy of the interferometry system measurements candegrade over the system's lifetime. Accordingly, the surface figures ofthe mirrors used in a metrology system should be re-measuredperiodically to maintain system accuracy.

Accurate knowledge of the surface figure of a plane mirror measurementobject can be particularly beneficial where metrology systems are usedin applications with high accuracy requirements. An example of such anapplication is in lithography tools where metrology systems are used tomonitor the position of a stage that supports a wafer or a mask withinthe tool.

In some embodiments, ex situ measurement methods can be used todetermine information about the surface figure of a mirror. In thesemethods, the mirror is removed from the lithography tool and measuredusing another piece of apparatus. However, the metrology system cannotbe used until the mirror is replaced, so the tool is generallyunproductive during such maintenance. Furthermore, the surface figure ofa mirror can change when the tool is reinserted into the tool (e.g., asa result of stresses associated with the mechanical attachment of themirror to the tool), introducing unaccounted sources of error into themetrology system.

In situ measurement methods can mitigate these errors because thesurface figure is measured while it is attached to the tool, after themirror has adapted to stresses associated with its attachment to thetool. Moreover, in situ methods can improve tool throughput by reducingthe amount of time a tool is offline for servicing. In this disclosure,interferometry systems and methods are discussed in which informationabout a surface figure of a mirror can be obtained in a lithography toolduring the operation of the tool or while the tool is offline. Moregenerally, the systems and methods are not limited to use in lithographytools, and can be used in other applications as well (e.g., in beamwriting systems).

In embodiments, procedures for determining information about a surfacefigure of a stage mirror includes measuring values of a seconddifference parameter (SDP) for the mirror. The SDP of a mirror can beexpressed as a series of orthogonal basis functions (e.g., a Fourierseries), where the coefficients (e.g., Fourier coefficients) of theseries are related by a transfer function to corresponding coefficientsof a series representation of a mirror surface figure function, ξ.

However, the SDP function is not necessarily mathematically invertibleto obtain a complete set of orthogonal basis functions used in a seriesrepresentation. Accordingly, additional functions are defined that,together with the SDP, allow use of a set of orthogonal basis functionswhich permit inversion of the SDP to a conjugate (e.g., spatialfrequency) domain. These additional functions are referred to asextended SDP related parameters (SDP^(e)s).

SDP and SDP^(e)s will generally have a reduced or nominally zerosensitivity to certain spatial frequency components of a spectralrepresentation of a surface figure where the certain spatial frequencycomponents are related to the axis spacing of the interferometers usedto measure data for the SDP and SDP^(e)s. In certain embodiments,information about the surface figure at these spatial frequencycomponents can be acquired using supplementary methods, and thesupplemental information can be used used to improve the accuracy of thesurface figure determined from the SDP and SDP^(e)s.

Information about the certain spatial frequency components of the mirrorsurface figure can be obtained from sets of measurements in situ of analignment wafer containing one or more nominally parallel linear arraysof alignment marks which are parallel to the surface of the stagemirror. The spacings of the alignment marks in the linear arrays arerelated to the spatial frequencies of components where the principlemethods involving SDP and SDP^(e)s have reduced or nominally zerosensitivity.

In some embodiments, a set of measurements can be acquired correspondingto a first measurement of the positions of the alignment marks of thealignment wafer in an original position and a second measurement ofpositions of the alignment marks of the alignment wafer in a displacedposition relative to the original position by a displacement related toa spatial frequency of components where the principle methods involvingSDP and SDP^(e)s have reduced or nominally zero sensitivity.

In certain embodiments, information about the spatial frequencycomponents where the principle methods have reduced or nominally zerosensitivity is based on sets of measurements in situ of an alignmentwafer containing one or more nominally parallel linear arrays ofalignment marks which are parallel to the surface of the stage mirror.For these procedures, a set of measurements corresponds to a firstmeasurement of the positions of the alignment marks of the alignmentwafer in an original orientation and a second measurement of positionsof the alignment marks of the alignment wafer rotated by 180 degrees.

The values of SDP and SDP^(e)s can be measured in situ as a paralleloperation with or as a serial operation with the normal processing cycleof wafers, in situ during periods other than during a normal processingcycle of wafers such as during a programmed maintenance procedure of alithography tool, or ex-situ prior to installation of respective stagemirrors in a lithography tool. Cyclic errors that are present in thelinear displacement measurements are eliminated and/or compensated byuse of one of more procedures. Improved statistical accuracy in measuredvalues of SDP and the other SDP related parameters is obtained by takingadvantage of the relatively low bandwidth of measured values of SDP andthe other SDP related parameters compared to the bandwidth of thecorresponding linear displacement measurements using averaging or lowpass filtering. Local spatial derivatives of the SDP and the other SDPrelated parameters are also measured using measured values of SDP andthe other SDP related parameters.

There may be offset errors in SDP and the other SDP related parametersbecause SDP and the other SDP related parameters are derived from threedifferent interferometer measurements where each interferometer measuresonly relative changes between respective reference and measurement beampaths. In addition, the offset errors may change with time because therelative path lengths of the measurement and reference beams for each ofthe three interferometers may change with respect to each other, e.g.due to changes in temperature. The effects of the offset errors as wellas the effects of a stage rotation or rotations that may occur duringthe acquisition of measured values of SDP and the other SDP relatedparameters on subsequently determined stage mirror surface figurefunctions can be eliminated by use of properties of SDP relative to theother SDP related parameters. The angle or angles between the x-axis andy-axis stage mirrors appropriate to surface figure functions in one ortwo different x-y planes are measured by use of an alignment wafer andthe alignment wafer rotated by 90 degrees. The two different planescorrespond to two different planes of an interferometer systemcomprising two different multiple-axes/plane interferometers.

Various aspects of the invention are summarized as follows.

In general, in a first aspect, the invention features methods thatinclude locating a plurality of alignment marks on a moveable stage,interferometrically measuring a position of a measurement object alongan interferometer axis for each of the alignment mark locations, andusing the interferometric position measurements to derive informationabout a surface figure of the measurement object. The position of themeasurement object is measured using an interferometry assembly andeither the measurement object or the interferometry assembly areattached to the stage.

Implementations of the methods can include one or more of the followingfeatures and/or features of other aspects. For example, The informationcan include information related to a certain spatial frequency componentof the surface figure of the measurement object.

The method can include using the information to determine the surfacefigure of the measurement object. The surface figure of the measurementobject can be determined based on values of a parameter associated witha displacement of the measurement object along three differentmeasurement axes in addition to the information. The parameter valuescan be determined by interferometrically monitoring the displacement ofthe measurement object along each of the three different interferometeraxes while moving the measurement object relative to the interferometryassembly, and determining the parameter values for different positionsof the measurement object from the monitored displacements, wherein fora given position the parameter is based on the displacements of themeasurement object along each of the three different interferometer axesat the given position.

The surface figure of the measurement object can be determined based ona frequency transform of the parameter values. The frequency transformcan be a Fourier transform. The information can include informationrelated to a certain spatial frequency component of the surface figureof the measurement object at which the frequency transform of theparameter values is substantially insensitive. The parameter can be asecond difference parameter.

The plurality of alignment marks can include a linear array of alignmentmarks. The linear array of the alignment marks can be nominally parallelto the measurement object during the measuring. In some embodiments,adjacent alignment marks in the linear array are separated by adistance, d_(a), and the information about the surface figure of themeasurement object is related to a spatial frequency component of thesurface figure having a spatial wavelength Λ greater than d_(a). Incertain embodiments, Λ=4d_(a).

The alignment marks can be located on the surface of an object supportedby the moveable stage, and locating the alignment marks includeslocating the alignment marks for at least two different positions of theobject. The two different object positions can include a first positionand a second position where the object is rotated 180° with respect tothe first position. Alternatively, or additionally, the two differentobject positions can include a first position and a second positionwhere the object is translated relative to the first position. Thetranslation can be by an amount related to a spacing of the alignmentmarks.

The method can include using the information about the surface figure ofthe measurement object to improve the accuracy of measurements madeusing the interferometry assembly and the measurement object.

In some embodiments, the method includes using a lithography tool toexpose a substrate supported by the moveable stage with a radiationpattern while interferometrically monitoring a distance between theinterferometry assembly and the measurement object, wherein the positionof the substrate relative to a reference frame is related to thedistance between the interferometry assembly and the measurement object.

In another aspect, the invention features lithography methods for use infabricating integrated circuits on a wafer. The lithography methodsinclude supporting the wafer on a moveable stage, imaging spatiallypatterned radiation onto the wafer, adjusting the position of the stage,and monitoring the position of the stage using a measurement object andinformation about the surface figure of the measurement object derivedusing methods discussed in other aspects of the invention to improve theaccuracy of the monitored position of the stage.

In another aspect, the invention features lithography methods for use inthe fabrication of integrated circuits that include directing inputradiation through a mask to produce spatially patterned radiation,positioning the mask relative to the input radiation, monitoring theposition of the mask relative to the input radiation using a measurementobject and information about the surface figure of the measurementobject derived using methods discussed in other aspects of the inventionto improve the accuracy of the monitored position of the mask, andimaging the spatially patterned radiation onto a wafer.

In a further aspect, the invention features lithography methods forfabricating integrated circuits on a wafer that include positioning afirst component of a lithography system relative to a second componentof a lithography system to expose the wafer to spatially patternedradiation, and monitoring the position of the first component relativeto the second component using a measurement object and using informationabout the surface figure of the measurement object derived using methodsdiscussed in other aspects of the invention to improve the accuracy ofthe monitored position of the first component.

In a further aspect, the invention features methods for fabricatingintegrated circuits that include applying a resist to a wafer,

forming a pattern of a mask in the resist by exposing the wafer toradiation using the lithography methods discussed in other aspects ofthe invention, and producing an integrated circuit from the wafer.

In general, in another aspect, the invention features systems thatinclude a moveable stage, an alignment sensor configured to locatealignment marks associated with the moveable stage,

a interferometer assembly configured to produce three output beams eachincluding interferometric information about a distance between theinterferometer and a measurement object along a respective axis, theinterferometer assembly or the measurement object being attached to themoveable stage, and an electronic processor configured to deriveinformation about a surface figure of the measurement object based ondata acquired by locating the plurality of alignment marks with thealignment sensor and measuring the position of the measurement objectalong one of the respective axes of the interferometer assembly for eachof the alignment mark locations.

Embodiments of the system can include one or more of the followingfeatures and/or features of other aspects. For example, The measurementobject can be a plane mirror. A surface of the stage can include thealignment marks. Alternatively, or additionally, the stage can beconfigured to support an object that includes the alignment marks.

The alignment sensor can be an optical alignment sensor, such as amicroscope.

In another aspect, the invention features lithography systems for use infabricating integrated circuits on a wafer. The lithography systemsinclude the system disclosed above,

an illumination system for imaging spatially patterned radiation onto awafer supported by the moveable stage, and a positioning system foradjusting the position of the stage relative to the imaged radiation.The interferometer assembly is configured to monitor the position of thewafer relative to the imaged radiation and electronic processor isconfigured to use the information about the surface figure of themeasurement object to improve the accuracy of the monitored position ofthe wafer.

In a further aspect, the invention features methods for fabricatingintegrated circuits that include applying a resist to a wafer, forming apattern of a mask in the resist by exposing the wafer to radiation usingthe lithography system discussed above, and producing an integratedcircuit from the wafer.

In another aspect, the invention features beam writing systems for usein fabricating a lithography mask. The beam writing systems include thesystem discussed above, a source providing a write beam to pattern asubstrate supported by the moveable stage, a beam directing assembly fordelivering the write beam to the substrate, and a positioning system forpositioning the stage and beam directing assembly relative one another.The interferometer assembly is configured to monitor the position of thestage relative to the beam directing assembly and electronic processoris configured to use the information about the surface figure of themeasurement object to improve the accuracy of the monitored position ofthe stage.

In a further aspect, the invention features methods for fabricating alithography mask that include directing a beam to a substrate using thebeam writing system discussed above, varying the intensity or theposition of the beam at the substrate to form a pattern in thesubstrate, and forming the lithography mask from the patternedsubstrate.

Other aspects and advantages of the invention will be apparent from thedetailed description and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 a is a perspective drawing of an interferometer system comprisingtwo three-axis/plane interferometers.

FIG. 1 b is a diagram that shows the pattern of measurement beams frominterferometer system of FIG. 1 a at a stage mirror that serves asmeasurement object for interferometers of the interferometer system.

FIG. 2 a is a diagrammatic perspective view of an interferometer system.

FIG. 2 b is a diagram showing domains for a second difference parameter(SDP) and extended SDP-related parameters on a surface of a mirror.

FIG. 3 is a plot comparing the transfer function magnitude as a functionof spatial frequency for different values of η.

FIG. 4 is a is a diagram that shows a linear array of alignment marks onan alignment wafer.

FIG. 5 is a schematic diagram of an embodiment of a lithography toolthat includes an interferometer.

FIG. 6 a and FIG. 6 b are flow charts that describe steps for makingintegrated circuits.

FIG. 7 is a schematic of a beam writing system that includes aninterferometry system.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

FIG. 1 a shows an embodiment of a multi-axis/plane mirror interferometer100, which directs multiple measurement beams to each contact ameasurement object 120 (e.g., a plane mirror measurement object) twice.Interferometer 100 produces multiple output beams 181-183 and 191-193each including interferometric information about changes in distancebetween the interferometry system and the measurement object along acorresponding measurement axis.

Interferometer 100 has the property that the output beams each includesa measurement component that makes one pass to the measurement objectalong one of two common measurement beam paths before being directedalong separate measurement beam paths for the second pass to themeasurement object. Similar interferometers are disclosed in commonlyowned U.S. patent application Ser. No. 10/351,707 by Henry, A. Hillfiled Jan. 27, 2003 and entitled “MULTIPLE DEGREE OF FREEDOMINTERFEROMETER,” the contents of which are incorporated herein byreference.

Interferometer 100 includes a non-polarizing beam splitter 110, whichsplits a primary input beam 111 into two secondary input beams 112A and112B. Interferometer 100 also includes a polarizing beam splitter 115,which splits secondary input beams 112A and 112B into primarymeasurement beams 113A and 113B, and primary reference beams 114A and114B, respectively. Interferometer 100 directs primary measurement beams113A and 113B along paths that contact measurement object 120 atdifferent locations in a vertical direction. Similarly, primaryreference beams 114A and 114B are directed along reference beam pathsthat contact a reference mirror 130 at different locations.Interferometer 100 also includes quarter wave plates 122 and 124.Quarter wave plate 122 is located between polarizing beam splitter 115and measurement object 120, while quarter wave plate 124 is locatedbetween polarizing beam splitter 115 and the reference mirror. Thequarter wave plates rotate by 90° the polarization state of doublepassed beams directed between the polarizing beam splitter and themeasurement object or reference mirror. Accordingly, the polarizing beamsplitter transmits an incoming beam that would have been reflected inits out-going polarization state.

The following description pertains to primary measurement beam 113A andprimary reference beam 114A. Interferometer 100 directs measurement beam113B and reference beam 114B along analogous paths. Polarizing beamsplifter (PBS) 115 transmits reflected primary measurement beam 113B,which is reflected back towards PBS 115 by a retroreflector 140 (asimilar retroreflector 141 reflects primary measurement beam 113B). Acompound optical component 150 including non-polarizing beam splitters151 and 152 and reflector 153 split primary measurement beam 113A intothree secondary measurement beams 161, 162, and 163. PBS 115 transmitsthe three secondary measurement beams, which propagate along paths thatcontact measurement object 120 at three different positions in ahorizontal plane shared by primary measurement beam 113A. PBS 115 thendirects the three secondary measurement beams reflected from measurementobject 120 along output path. Similarly, compound optical component 150splits primary measurement beam 113B into three secondary measurementbeams 171, 172, and 173 which propagate along paths that contactmeasurement object 120 at three different positions in a horizontalplane shared by primary measurement beam 113B.

PBS 115 reflects primary reference beam 114A towards retroreflector 140.As for the primary measurement beam, optical component 150 splitsprimary reference beam 114A reflected by retroreflector 140 into threesecondary reference beams. PBS 115 reflects the secondary referencebeams towards reference mirror 130 along paths at three differentpositions in a plane shared by primary reference beam 114A. PBS 115transmits the secondary reference beams reflected from reference object130 along output paths so that they overlap with measurement beams 161,162, and 163 to form output beams 181, 182, and 183, respectively. Thephase of the output beams carries information about the position of themeasurement object along three measurement axes defined by the primarymeasurement beam's path and the secondary measurement beams' paths.

Interferometer 100 also includes a window 160 located between quarterwave plate 122 and measurement object 120.

The pattern of measurement beams incident on a plane mirror measurementobject is shown in FIG. 1 b. The angle of incidence of measurement beamsat the mirror surface is nominally zero when the measurement axes areparallel to the x-axis of a coordinate system. The locations of themeasurement axes of the top multiple-axis/plane interferometercorresponding to x₁, x₂, and x₃ are shown in FIG. 1 b. The spacingsbetween measurement axes corresponding to x₁ and x₂ and to x₁ and x₃ areb₂ and b₃, respectively. In general, b₂ and b₃ can vary as desired. b₂can be the same as or different from (b₃−b₂). In some embodiments, theaxis spacing can be relatively narrow (e.g., about 10 cm or less, about5 cm or less, about 3 cm or less, about 2 cm or less). For example,where the resolution of a measurement depends on the spacing the axes,having relatively narrow spacing between at least two of the measurementaxes can provide increased sensitivity to higher frequencies in ameasurement.

Also shown in FIG. 1 b is the location corresponding to the primarysingle pass measurement beam x′₀ and the locations corresponding to thesecond pass measurement beams x′₁, x′₂, and x′₃. The relationshipbetween a linear displacement measurement corresponding to a double passto the stage mirror and the linear displacement measurementscorresponding to a single pass to the stage mirror is given by

$\begin{matrix}{{x_{j} = {\frac{1}{2}\left( {x_{j}^{\prime} + x_{0}^{\prime}} \right)}},{j = 1},2,{{and}\mspace{14mu} 3.}} & (2)\end{matrix}$

The difference between two linear displacements x_(i) and x_(j), i≠j,referred to as a first difference parameter (FDP) is independent of x′₀,i.e.,

$\begin{matrix}{{{x_{i} - x_{j}} = {\frac{1}{2}\left( {x_{i}^{\prime} - x_{j}^{\prime}} \right)}},i,{j = 1},2,{{and}{\mspace{11mu}\;}3},{i \neq {j.}}} & (3)\end{matrix}$

Reference is made to FIG. 2 a which is a diagrammatic perspective viewof an interferometry system 15 that employs a pair of orthogonallyarranged interferometers or interferometer subsystems by which the shapeof on-stage mounted stage mirrors may be characterized in situ along oneor more datum lines. As shown in FIG. 2 a, system 15 includes a stage 16that can form part of a microlithography tool. Affixed to stage 16 is aplane stage mirror 50 having a y-z reflective surface 51 elongated inthe y-direction.

Also, affixed to stage 16 is another plane stage mirror 60 having an x-zreflective surface 61 elongated in the x-direction. Mirrors 50 and 60are mounted on stage 16 so that their reflective surfaces, 51 and 61,respectively, are nominally orthogonal to one another. Stage 16 isotherwise mounted for translations nominally in the x-y plane but mayexperience small angular rotations about the x, y, and z axes due to,e.g., bearing and drive mechanism tolerances. In normal operation,system 15 is adapted to be operated for scanning in the y-direction forset values of x.

Fixedly mounted off-stage (e.g., to a frame of the lithography tool) isan interferometer (or interferometer subsystem) that is generallyindicated at 10. Interferometer 10 is configured to monitor the positionof reflecting surface 51 along three measurement axes (e.g., x₁, x₂, andx₃) in each of two planes parallel to the x-axis, providing a measure ofthe position of stage 16 in the x-direction and the angular rotations ofstage 16 about the y- and z-axes as stage 16 translates in they-direction. Interferometer 10 includes two three-axis/planeinterferometers such as interferometer 100 shown in FIG. 1 a andarranged so that interferometric beams travel to and from mirror 50generally along optical paths designated generally as path 12.

Also fixedly mounted off-stage (e.g., to the frame of the lithographytool) is an interferometer (or interferometer subsystem) that isgenerally indicated at 20. Interferometer 20 can be used to monitor theposition of reflecting surface 61 along three measurement axes (e.g.,y₁, y₂, and y₃) in each of two planes parallel to the y-axis, providinga measure of the position of stage 16 in the y-direction and the angularrotations of stage 16 about the x- and z-axes as stage 16 translates inthe x-direction. Interferometer 20 includes two three-axis/planeinterferometers such as interferometer 100 shown in FIG. 1 a andarranged so that interferometric beams travel to and from mirror 60generally along an optical path designated as 22.

Also shown in FIG. 2 a is an alignment sensor 14 fixedly mountedoff-stage and arranged to locate alignment marks at axis 18 that are ona surface of stage 16 or on a wafer supported by stage 16. Alignmentsensor 14 can be an optical sensor, such as a microscope (e.g.,including a camera to monitor a field on the surface of stage 16 at axis18). Typically, the alignment marks are protrusions or recesses in asurface that are optically distinguishable from their surroundings.

In general, simultaneously sampled values for x₁, x₂, and x₃ for x-axisstage mirror 50 are acquired as a function of position of the mirror inthe y-direction with the corresponding x-axis location and the stagemirror orientation nominally held at fixed values. In addition, valuesfor y₁, y₂, and y₃ for y-axis stage mirror 60 are measured as a functionof the position in the x-direction of the y-axis stage mirror with thecorresponding y-axis location and stage orientation nominally held atfixed values. The orientation of the stage can be determined initiallyfrom information obtained using one or both of the three-axis/planeinterferometers.

The SDP and the SDP^(e) values for the x-axis stage mirror arecalculated as a function of position of the x-axis stage mirror in they-direction with the corresponding x-axis location and the stage mirrororientation nominally held at fixed values from the data acquired asdiscussed previously. Also the SDP and the SDP^(e) values for the y-axisstage mirror are measured as a function of position in the x-directionof the y-axis stage mirror with the corresponding y-axis location andstage orientation nominally held at fixed values. Increased sensitivityto high spatial frequency components of the surface figure of a stagemirror can be obtained by measuring the respective SDP and the SDP^(e)values with the stage oriented at large yaw angles and/or largemeasurement path lengths to the stage mirror, i.e., for largemeasurement beam shears at the respective measuring three-axis/planeinterferometer.

The measurements of the respective SDP values for the x-axis and y-axisstage mirrors do not require monitoring of changes in stage orientationduring the respective scanning of the stage mirrors other than tomaintain the stage at a fixed nominal value since SDP is independent ofstage mirror orientation except for third order effects. However,accurate monitoring of changes in stage orientation during scanning of astage mirror may be required when a three-axis/plane interferometer isused to measure SDP^(e)s since these parameters may be sensitive tochanges in stage orientation.

The surface figure for an x-axis stage mirror will, in general, be afunction of both y and z and the surface figure for the y-axis stagemirror will, in general, be a function of both x and z. The z dependenceof surface figures is suppressed in the remaining description of thedata processing algorithms except where explicitly noted. Thegeneralization to cover the z dependent properties can be addressed byusing two three-axis/plane interferometer system in conjunction with aprocedure to measure the angle between the x-axis and y-axis stagemirrors in two different parallel planes displaced along the z-directioncorresponding to the two planes defined by the two three-axis/planeinterferometer system.

The following description is given in terms of procedures used withrespect to the x-axis stage mirror since the corresponding algorithmsused with respect to the y-axis stage mirror can be easily obtained by atransformation of indices. Once the system has acquired values forx₁(y), x₂(y), and x₃(y) for one or more locations of the stage withrespect to the x-axis, values for SDP and on or more SDP^(e)s arecalculated from which ξ(y) can be determined.

Turning now to mathematical expressions for the SDP and SDP^(e), SDP canbe defined for a three-axis/plane interferometer such that it is notsensitive to either a displacement of a respective mirror or to therotation of the mirror except through a third and/or higher order effectinvolving the angle of stage rotation in a plane defined by thethree-axis/plane interferometer and departures of the stage mirrorsurface from a plane.

Different combinations of displacement measurements x₁, x₂, and x₃ maybe used in the definition of a SDP. One definition of a SDP for anx-axis stage mirror is for example

$\begin{matrix}{{{{SDP}(y)} \equiv {\left( {x_{2} - x_{1}} \right) - {\frac{b_{2}}{b_{3} - b_{2}}\left( {x_{3} - x_{2}} \right)}}}{or}} & (4) \\{{{{SDP}(y)} = {\left( {x_{2} - x_{1}} \right) - {\eta\left( {x_{3} - x_{2}} \right)}}}{where}} & (5) \\{\eta \equiv {\frac{b_{2}}{b_{3} - b_{2}}.}} & (6)\end{matrix}$Note that SDP can be written in terms of single pass displacements usingEquation (3), i.e.,

$\begin{matrix}{{{SDP}(y)} = {{\frac{1}{2}\left\lbrack {\left( {x_{2}^{\prime} - x_{1}^{\prime}} \right) - {\eta\left( {x_{3}^{\prime} - x_{2}^{\prime}} \right)}} \right\rbrack}.}} & (7)\end{matrix}$

Of course, corresponding equations apply for a y-axis stage mirror.

The lowest order term in a series representation of the SDP is relatedto the second spatial derivative of a functional representation of thesurface figure (e.g., corresponding to the second order term of a Taylorseries representation of a surface figure). Accordingly, for relativelylarge spatial wavelengths, the SDP is substantially equal to the secondderivative of the mirror surface figure. Generally, the SDP isdetermined for a line intersecting a mirror in a plane determined by theorientation of an interferometry assembly with respect to the stagemirror.

Certain properties of the three-axis/plane interferometer relevant tomeasurement of values of SDP are apparent. For example, SDP isindependent of a displacement of the mirror for which SDP is beingmeasured. Furthermore, SDP is generally independent of a rotation of themirror for which SDP is being measured except through a third order orhigher effects. SDP should be independent of properties of the primarysingle pass measurement beam path x′₀ in the three-axis/planeinterferometer. SDP should be independent of properties of theretroreflector in the three-axis/plane interferometer. Furthermore, SDPshould be independent of changes in the average temperature of thethree-axis/plane interferometer and should be independent of lineartemperature gradients in the three-axis/plane interferometer. SDP shouldbe independent of linear spatial gradients in the refractive indices ofcertain components in a three-axis/plane interferometer. SDP should beindependent of linear spatial gradients in the refractive indices and/orthickness of cements between components in a three-axis/planeinterferometer. SDP should also be independent of “prism effects”introduced in the manufacture of components of a three-axis/planeinterferometer used to measure the SDP.

SDP^(e)s are designed to exhibit spatial filtering properties of thesurface figure ξ that are similar to the spatial filtering properties ofthe surface figure ξ by SDP for a given portion of the respective stagemirror. SDP^(e)s can also be defined that exhibit invariance propertiessimilar to those of SDP in procedures for determining the surface figureξ. As a consequence of the invariance properties, neither knowledge ofeither the orientation of the stage mirror nor accurate knowledge ofposition of the stage mirror is required, e.g., accurate knowledge isnot required of the position in the direction of the displacementmeasurements used to compute the SDP^(e)s. However, changes in theorientation of the stage mirror are monitored during the acquisition oflinear displacement measurements used to determine SDP^(e)s.

SDP^(e)s can be defined such that the properties of the transferfunctions with respect to the surface figure ξ is similar to theproperties of the transfer function of SDP with respect to the surfacefigure ξ. The properties of SDP^(e)s with respect to terms of linearfirst pass displacement measurements is developed from knowledge of theproperties of the SDP with respect to terms of linear first passdisplacement measurements for a surface error function ξ that isperiodic with a fundamental periodicity length L and for domains in ythat are exclusive of the domain in y for which the corresponding SDP isvalid. Two SDP^(e)s, denoted as SDP₁ ^(e) and SDP₂ ^(e), for a domaingiven byL=q2b₃, q=2,3, . . . ,  (8)

are given by the formulae

$\begin{matrix}{{{{SDP}_{1}^{e}\left( {y,\vartheta_{z}} \right)} \equiv {{\frac{1}{2}\begin{Bmatrix}\left\lbrack {{x_{2}^{\prime}(y)} - {x_{1}^{\prime}(y)}} \right\rbrack \\{+ {\eta\left\lbrack {{x_{2}^{\prime}(y)} - {x_{1}^{\prime}\left( {y - L + {2b_{3}}} \right)}} \right\rbrack}}\end{Bmatrix}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{\frac{1}{2} - \frac{2\left( {b_{3} - b_{2}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},} & (9) \\{{{{SDP}_{2}^{e}\left( {y,\vartheta_{z}} \right)} \equiv {{{- \frac{1}{2}}\left\{ {\left\lbrack {{x_{3}^{\prime}\left( {y + L - {2b_{3}}} \right)} - {x_{2}^{\prime}(y)}} \right\rbrack + {\eta\left\lbrack {{x_{3}^{\prime}(y)} - {x_{2}^{\prime}(y)}} \right\rbrack}} \right\}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + {\frac{2b_{2}}{L}.}}}} & (10)\end{matrix}$where the last term in Equations (9) and (10) is a third order term withan origin in a second order geometric term such as described in commonlyowned U.S. patent application Ser. No. 10/347,271 entitled “COMPENSATIONFOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRRORINTERFEROMETERS” and U.S. patent application Ser. No. 10/872,304entitled “COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS INPLANE MIRROR INTERFEROMETER METROLOGY SYSTEMS,” both of which are byHenry A. Hill and both of which are incorporated herein in theirentirety by reference.

Expressing SDP₁ ^(e) and SDP₂ ^(e) in terms of pairs of single passdisplacement measurements obtained at time t_(j), j=1, 2, . . . , thefollowing equations are derived from Equations (9) and (10):

$\begin{matrix}{{{{SDP}_{1}^{e}\left( {y,\vartheta_{z}} \right)} \equiv {{\frac{1}{2}\begin{matrix}{\left( {1 + \eta} \right)\left\lbrack {{x_{2}^{\prime}\left( {y,t_{1}} \right)} - {x_{1}^{\prime}\left( {y,t_{1}} \right)}} \right\rbrack} \\{{+ \eta}\begin{Bmatrix}\left\lbrack {{x_{3}^{\prime}\left( {{y - {2b_{3}}},t_{2}} \right)} - {x_{1}^{\prime}\left( {{y - {2b_{3}}},t_{2}} \right)}} \right\rbrack \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y - {{2 \times 2}b_{3}}},t_{3}} \right)} - {x_{1}^{\prime}\left( {{y - {{2 \times 2}b_{3}}},t_{3}} \right)}} \right\rbrack} \\\vdots \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y - {\left( {q - 1} \right)2\; b_{3}}},t_{q}} \right)} - {x_{1}^{\prime}\left( {{y - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)}} \right\rbrack}\end{Bmatrix}}\end{matrix}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{\frac{1}{2} - \frac{2\left( {b_{3} - b_{2}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},} & (11) \\{{{{SDP}_{2}^{e}\left( {y,\vartheta_{z}} \right)} = {{{- \frac{1}{2}}\begin{matrix}{\left( {1 + \eta} \right)\left\lbrack {{x_{3}^{\prime}\left( {y,t_{1}} \right)} - {x_{2}^{\prime}\left( {y,t_{1}} \right)}} \right\rbrack} \\{+ \begin{Bmatrix}{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {2b_{3}}},t_{2}} \right)} - {x_{1}^{\prime}\left( {{y + L - {2b_{3}}},t_{2}} \right)}} \right\rbrack} \\\begin{matrix}{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {{2 \times 2}b_{3}}},t_{3}} \right)} -} \right.} \\\left. {x_{1}^{\prime}\left( {{y + L - {{2 \times 2}b_{3}}},t_{3}} \right)} \right\rbrack\end{matrix} \\\vdots \\\begin{matrix}{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)} -} \right.} \\\left. {x_{1}^{\prime}\left( {{y + L - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)} \right\rbrack\end{matrix}\end{Bmatrix}}\end{matrix}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + \frac{2b_{2}}{L}}},{{{where}\mspace{14mu} t_{i}} \neq t_{j}},{i \neq {j.}}} & (12)\end{matrix}$

FIG. 2 b illustrate the relationship of the domains for which SDP, SDP₁^(e), and SDP₂ ^(e) are defined on mirror surface 51. Domain Lcorresponds to a portion of mirror surface 51 along a scan line 201 inthe y-direction. The locations of the first pass beams on mirror surface51 are shown for the mirror at position y_(i). Note that the domains iny for SDP, SDP₁ ^(e), and SDP₂ ^(e), i.e.,

${{{- \frac{1}{2}} + \left( \frac{2b_{2}}{L} \right)} \leq \frac{y}{L} \leq {\frac{1}{2} - \left( \frac{{2b_{3}} - {2b_{2}}}{L} \right)}},{{\frac{1}{2} - \frac{\left( {{2b_{3}} - {2b_{2}}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},{{{and}{\;\mspace{11mu}} - \frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + \frac{2b_{2}}{L}}},$are mutually exclusive and that the combined domains in y of the threedomains cover the domain

${- \frac{1}{2}} \leq \frac{y}{L} \leq {\frac{1}{2}.}$

For a section of the x-axis stage mirror covering domain L in y, thesurface figure is represented mathematically as a function ξ(y) whichcan be expressed by the Fourier series

$\begin{matrix}{{{\xi(y)} = {{\sum\limits_{m = 0}^{N}{A_{m}{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}}} + {\sum\limits_{m = 1}^{N}{B_{m}{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq \frac{1}{2}},} & (13)\end{matrix}$where N is an integer determined by consideration of the spatialfrequencies that are to be included in the series representation. Theterm represented by A₀ which is sometimes referred to as a “piston” typeerror is included in Equation (13) for completeness. An error of thistype is equivalent to the effect of a displacement of the stage mirrorin the direction orthogonal to the stage mirror surface and as such isnot considered an intrinsic property of the surface figure. Using thedefinition of SDP given by Equation (7), the corresponding series forSDP is next written as

$\begin{matrix}{{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}{{\cos\left( {m\; 2\pi\frac{y}{L}} \right)} \times \begin{Bmatrix}{{A_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\pi\;\frac{2b_{2}}{L}} \right)} - {\eta\;{\cos\left( {m\; 2\;\pi\frac{{2\; b_{3}} - {2b_{2}}}{L}} \right)}}} \right\rbrack}\;} \\{+ {B_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{2\; b_{2}}{L}} \right)} - {\eta\;{\sin\left( {m\; 2\;\pi\frac{{2\; b_{3}} - {2b_{2}}}{L}} \right)}}} \right\rbrack}}\end{Bmatrix}}}} + {\frac{1}{2}{\sum\limits_{m = 1}^{N}{{\sin\left( {m\; 2\pi\frac{y}{L}} \right)} \times \begin{Bmatrix}{- {A_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\;\frac{2b_{2}}{L}} \right)} - {\eta\;{\sin\left( {m\; 2\;\pi\frac{{2b_{3}} - {2b_{2}}}{L}} \right)}}} \right\rbrack}} \\{+ {B_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{2\; b_{2}}{L}} \right)} - {\eta\;{\cos\left( {m\; 2\pi\frac{{2b_{3}} - {2b_{2}}}{L}}\; \right)}}} \right\rbrack}}\end{Bmatrix}}}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{{- \frac{1}{2}}\left( {1 - \frac{2b_{2}}{L}} \right)} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - \left( \frac{{2b_{3}} - {2b_{2}}}{L} \right)} \right\rbrack}},} & (14)\end{matrix}$

where x is a linear displacement of the stage mirror based on one ormore of the linear displacements x_(i), i=1,2, and/or 3, and Θ_(z) isthe angular orientation of the state mirror in the x-y plane. The lastterm in Equation (14) is a third order term with an origin in a secondorder geometric term such as described in commonly owned U.S. patentapplication Ser. No. 10/347,271 entitled “COMPENSATION FOR GEOMETRICEFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERS” and U.S.patent application Ser. No. 10/872,304 entitled “COMPENSATION FORGEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERMETROLOGY SYSTEMS,” both of which are by Henry A. Hill and both of whichare incorporated herein in their entirety by reference.

The presence of the third order term in Equation (14) makes it possibleto measure the high spatial frequency components of ξ(y) in the x-yplane with increased sensitivity compared to the sensitivity representedby the first order, i.e., remaining, terms in Equation (14). The thirdorder term also makes it possible to obtain information aboutintermediate and low spatial frequency components of ξ in the x-z planecomplimentary to that obtained by the first order terms in Equation(14).

Note in Equation (14) that the constant term A₀ is not present. Thisproperty is subsequently used in a procedure for elimination of effectsof the offset errors and changes in stage orientation. The loss ofinformation about the constant term A₀ is not relevant to thedetermination of the intrinsic portion of the surface figure ξ since asnoted above, A₀ corresponds to a displacement of the stage mirror in thedirection orthogonal to the surface of the stage mirror.

Equation (14) may be rewritten in terms of η and b₃ and eliminating b₂by using Equation (6) with the result

$\begin{matrix}{{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}{{\cos\left( {m\; 2\pi\frac{y}{L}} \right)} \times \begin{Bmatrix}{\begin{matrix}{A_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\pi\frac{\eta}{1 + \eta}\;\frac{2b_{3}}{L}} \right)} -} \right.} \\\left. {\eta\;\cos\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2\; b_{3}}{L}} \right)} \right\rbrack\end{matrix}\;} \\{+ {B_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2\; b_{3}}{L}} \right)} - {\eta\;{\sin\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2\; b_{3}}{L}} \right)}}} \right\rbrack}}\end{Bmatrix}}}} + {\frac{1}{2}{\sum\limits_{m = 1}^{N}{{\sin\left( {m\; 2\pi\frac{y}{L}} \right)} \times \begin{Bmatrix}{- {A_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2\; b_{3}}{L}} \right)} - {\eta\;{\sin\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)}}} \right\rbrack}} \\\begin{matrix}{+ {B_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2\; b_{3}}{L}} \right)}} \right.}} \\\left. {{- \eta}\;{\cos\left( {m\; 2\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}}\; \right)}} \right\rbrack\end{matrix}\end{Bmatrix}}}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{- {\frac{1}{2}\left\lbrack {1 - {\left( \frac{\eta}{1 + \eta} \right)\frac{2\; b_{3}}{L}}} \right\rbrack}} \leq \frac{y}{L} \leq {{\frac{1}{2}\left\lbrack {1 - {\left( \frac{1}{1 + \eta} \right)\frac{2b_{3}}{L}}} \right\rbrack}.}}} & (15)\end{matrix}$

A contracted form of Equation (15) is obtained with the introduction ofa complex transfer function T(m) having real and imaginary amplitudes ofT_(Re) and T_(Im), respectively, as

$\begin{matrix}{{{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\;\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{- {\frac{1}{2}\left\lbrack {1 - {\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - {\left( \frac{1}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}}}{where}} & (16) \\{{A_{m}^{\prime} = {{A_{m}T_{Re}} + {B_{m}T_{Im}}}}{and}} & (17) \\{B_{m}^{\prime} = {{{- A_{m}}T_{Im}} + {B_{m}{T_{Re}.{with}}}}} & (18) \\{{T_{Re} = \left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} - {\eta\;{\cos\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)}}} \right\rbrack},{and}} & (19) \\{T_{Im} = {\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} - {\eta\;{\sin\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)}}} \right\rbrack.}} & (20)\end{matrix}$

The transfer function, T(m), relates the Fourier coefficients A′_(m) andB′_(m) of the SDP to the Fourier coefficients of ξ(y), A_(m) and B_(m).As is evident from Equations (19) and (20), T(m) depends on η and b₃/Lin addition to depending on spatial frequency (as indicated bydependence on m). The dependence on spatial frequency manifests as avarying sensitivity of the transfer function to spatial frequency, withzero sensitivity occurring at certain spatial frequencies. Lack ofsensitivity at a spatial frequency means that the SDP parameter does notcontain any information of the mirror surface for that frequency.Subsequent calculation of the function ξ(y) should make note of theselow sensitivity frequencies and handle them appropriately.

As an example, the magnitude |T(m)| of T(m) is plotted in FIG. 3 as afunction of m for L=4b₃ and 2b₃=50 nm and for η=7/4, 9/5, and 11/6. Thefirst zero in |T(m)| for m≧1, corresponding to zero sensitivity of thetransfer function, occurs at m=22, 28, and 34 for η=7/4, 9/5, and 11/6,respectively. The spatial wavelength Λ₁=4.55, 3.57, and 2.94 mm form=22, 28, and 34, respectively. The value of η is selected based onconsideration of the corresponding spatial wavelength Λ₁, whether or notthe displacements x_(i) are to be not sensitivity to spatial wavelengthΛ₁ corresponding to the first zero of |T(m)|, and the diameter of themeasurement beams.

There are values of η such as η=7/4, 9/5, and 11/6 belong to one set ofη's that lead to a property of measured values of x_(i) wherein thex_(i) are obtained using a corresponding three-axis/planeinterferometer. The first set of η's can be expressed as a ratio of twointegers that can not be reduced to a ratio of smaller integers and thenumerator is an odd integer. The second set of η's are a subset of thefirst set of η's wherein the denominator of the ratio is an eveninteger. An η of the first set of η's is defined, for example, by theratio

$\begin{matrix}{{\eta = \frac{{2n} - 1}{n}},{n = 2},3,\ldots} & (21)\end{matrix}$

An η of the corresponding second set of η's is a subset of the first setof η's defined by Equation (21) for even values of η.

A property associated with this first set of η's is that the effects ofthe spatial frequencies corresponding the first zero of |T(m)| for m>0,i.e., corresponding to the spatial frequency that is not measured,cancel out in the linear displacements x₁ and x₂.

A property associated with the second set of η's is that the effects ofthe spatial frequencies corresponding the first zero of |T(m)| for m>0cancel out in each of the linear displacements x₁, x₂, and x₃.

The first zero in |T(m)|for m≧1 and for the first set of η's is given bytwice the sum of the numerator and denominator of the ratio defining anη. For the η's defined by Equation (21), the corresponding values of mfor the first zero in |T(m)| for m>0 is given by the formulam=2(3n−1)  (22)

The spatial wavelength Λ₁ corresponding to the first zero in |T(m)| form≧1 is equal to 2b₃ divided by the sum of the numerator and denominatorof the ratio defining an η. For the η's defined by Equation (21), thecorresponding values of Λ₁ are

$\begin{matrix}{\Lambda_{1} = {\frac{2b_{3}}{{3n} - 1}.}} & (23)\end{matrix}$

The effects of the corresponding spatial frequency cancel out in x₁ andx₂ for η's of the first set of η's because the spacing b₂ between theprimary first pass measurement beam and the second pass measurementbeams of measurement axes x₁ and x₂ are an odd harmonic of Λ₁/2, i.e.,

$\begin{matrix}{b_{2} = {\left( {{2n} - 1} \right){\frac{\Lambda_{1}}{2}.}}} & (24)\end{matrix}$

The effects of the corresponding spatial frequency cancel out in x₁, x₂,and x₃ for η's of the second set of η's because the spacing b₂ betweenthe primary first pass measurement beam and the second pass measurementbeams of measurement axes x₁ and x₂ and the spacing (2b₃−b₂) between theprimary first pass measurement beam and the second pass measurement beamof measurement axis x₃ are each an odd harmonic of Λ₁/2, i.e., spacingb₂ is given by Equation (24) and spacing (2b₃−b₂) is given by theequation

$\begin{matrix}{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{4n} - 1} \right){\frac{\Lambda_{1}}{2}.}}} & (25)\end{matrix}$

There are other values of η's that belong to the first set which aregiven by the formulae

$\begin{matrix}{{\eta = \frac{{2n} + 1}{n}},{n\; = 2},3,\ldots} & (26)\end{matrix}$

The corresponding values for Λ₁, b₂, and (2b₃−b₂) are

$\begin{matrix}{\Lambda_{1} = \frac{2b_{3}}{{3n} + 1}} & (27) \\{{b_{2} = {\left( {{2n} + 1} \right)\frac{\Lambda_{1}}{2}}},} & (28) \\{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{4n} + 1} \right){\frac{\Lambda_{1}}{2}.}}} & (29)\end{matrix}$

Other values of η's that belong to the second set are given by theformulae

$\begin{matrix}{{\eta = \frac{{2n} \pm 1}{2n}},{n = 2},3,\ldots} & (30)\end{matrix}$

The corresponding values for Λ₁, b₂, and (2b₃−b₂) are

$\begin{matrix}{{\Lambda_{1} = \frac{2b_{3}}{{4n} \pm 1}},} & (31) \\{{b_{2} = {\left( {{2n} \pm 1} \right)\frac{\Lambda_{1}}{2}}},} & (32) \\{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{6n} \pm 1} \right){\frac{\Lambda_{1}}{2}.}}} & (33)\end{matrix}$

It was noted in the discussion associated with Equations (14), (15),(16), and (17) that the effects of the spatial frequency component witha spatial wavelength of Λ₁ cancel out in x₁, x₂, and x₃ for η's of thesecond set of η's. This is because the spacing b₂ between the primaryfirst pass measurement beam and the second pass measurement beams ofmeasurement axes x₁ and x₂ and the spacing (2b₃−b₂) between the primaryfirst pass measurement beam and the second pass measurement beam ofmeasurement axis x₃ are each an odd harmonic of Λ₁/2. However, thisadvantage may not be realized in certain end use applications as aresult of errors introduced in manufacturing the interferometer. Themanufacturing errors alter for example the locations of the measurementbeams which in turn introduce errors in η such that the condition ofEquation (13) may not be satisfied. When the effects of the spatialfrequency components with a spatial wavelength of Λ₁ and other similarspatial wavelengths do not cancel out in x₁, x₂, and x₃, the amplitudesof the spatial frequency components are measured in subsequentlydescribed procedures based on measurements of an alignment wafer. Theseprocedures will hereinafter be referred to as alignment waferprocedures.

The cut off spatial frequency in the determination of the surface figureproperties represented by ξ when using the three-axis/planeinterferometer will be determined by the spatial filtering properties ofmeasurement beams having a finite diameter, i.e., the finite size of ameasurement beam will serve as a low pass filter with respect to effectsof the higher frequency spatial frequencies of ξ. These effects areevaluated here for a measurement beam with a Gaussian profile. For aGaussian profile with a 1/e² diameter of 2s with respect to beamintensity, the effect of the spatial filtering of a Fourier seriescomponent with a spatial wavelength Λ is given by a transfer functionT_(Beam) where

$\begin{matrix}{T_{Beam} = {{\mathbb{e}}^{{- {(\frac{\pi^{2}}{8})}}{(\frac{2s}{\Lambda})}^{2}}.}} & (34)\end{matrix}$

Consider for example the effect of a Gaussian beam profile with a 1/e²diameter of 2s=5.0 mm. The attenuation effect of the spatial filteringwill be 0.225, 0.089, and 0.028 for Λ=4.55, 3.57, and 2.94 mm,respectively, for the respective Fourier series component amplitudes.

Using Equations (9) and (10), the series representations of SDP₁ ^(e)and SDP₂ ^(e) for the Fourier series representation of ξ(y) given byEquation (13) are

$\begin{matrix}{{{{SDP}_{1}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {{m2}\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} + {\eta\frac{L}{2}\vartheta_{z}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack} \leq \frac{y}{L} \leq \frac{1}{2}},} & (35) \\{{{{SDP}_{2}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} - {\frac{L}{2}\;\vartheta_{z}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq {- {{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}.}}}} & (36)\end{matrix}$

Note in Equations (35) and (36) that the constant term A₀ is notpresent. The discussion of this feature relevant to the intrinsicsurface figure function is the same as the corresponding portion of thediscussion related to the constant term A₀ not being included inEquation (14).

From a comparison of Equations (35) and (36), another property of SDP₁^(e) and SDP₂ ^(e) is evident, i.e., the respective contributions of theFourier series terms in A′_(m) and B′_(m) are the same in SDP₁ ^(e) andSDP₂ ^(e). Furthermore, the respective contributions of the Fourierseries terms in A′_(m) and B′_(m) in SDP₁ ^(e) and SDP₂ ^(e) are thesame as the respective contributions in SDP (see Equation (16)).

The SDP₁ ^(e) and SDP₂ ^(e) are classified as conjugate other SDPrelated parameters because of the cited property and because the firstorder terms ηLΘ_(z) and −LΘ_(z), respectively, multiplied by therespective domains in y, i.e., 2b₃/(1+η) and 2ηb₃/(1+η), respectively,are equal in magnitude and have opposite signs.

While certain SDP^(e)s have been explicitly defined, there are alsoother SDP^(e)s for L=4b₃+q2b₂+p2(b₃−b₂), q=1,2, . . . , p=1,2, . . . ,that exhibit similar properties as those of identified with respect toSDP₁ ^(e) and SDP₂ ^(e), such as an invariance to displacements of thestage mirror as evident on inspection of Equations (11) and (12).Another property of SDP₁ ^(e) and SDP₂ ^(e) is that the effect of astage rotation and the corresponding changes in SDP₁ ^(e) and SDP₂ ^(e)multiplied by their respective domain widths in y are equal in magnitudebut opposite in signs. This property is evident on examination ofEquations (35) and (36).

In some embodiments, the effects of the offset errors and the effects ofchanges in l_(z) that occur during the measurements of x₁(y), x₂(y), andx₃(y) are next included in the representations of SDP, SDP₁ ^(e), andSDP₂ ^(e). The offset errors for x₁, x₂, and x₃ are represented by E₁,E₂, and E₃, respectively. Offset errors arise in SDP because SDP isderived from three different interferometer measurements where each ofthe three interferometers can only measure relative changes in arespective reference and measurement beam paths. In addition, the offseterrors may change with time because the calibrations of each of thethree interferometers may change with respect to each other, e.g. due tochanges in temperature. The effects of changes in the orientation of thestage during the measurements of SDP, SDP₁ ^(e), and SDP₂ ^(e) areaccommodated by representing the orientation of the stage Θ_(z), asΘ_(z)(y).

The representations of SDP, SDP₁ ^(e) using Equation 11, and SDP₂ ^(e)using Equation 12, that include the effects of offset errors and changesin stage orientation are accordingly expressed as

$\begin{matrix}{{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} - {x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} - \left\lbrack {E_{1} - {\left( {1 + \eta} \right)E_{2}} + {\eta\; E_{3}}} \right\rbrack}},{{{- {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}};}} & (37) \\{{{{SDP}_{1}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\;\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} - {x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} - {\left( {1 + {\eta\; q}} \right)E_{1}} + {\left( {1 + \eta}\; \right)E_{2}} + {{\eta\left( {q - 1} \right)}E_{3}} + {\eta\left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right)} + {\eta\; b_{3}\begin{Bmatrix}{{\vartheta_{z}(y)} + {\vartheta_{z}\left( {y - {2b_{3}}} \right)}} \\{{+ \ldots} + {\vartheta_{z}\left\lbrack {y - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}}}},{{{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack} \leq \frac{y}{L} \leq \frac{1}{2}};{and}}} & (38) \\{{{{SDP}_{2}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\;\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} - {x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} + {\left( {q - 1} \right)E_{1}} + {\left( {1 + \eta} \right)E_{2}} - {\left( {q + \eta} \right)E_{3}} - \left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) + {b_{3}\begin{Bmatrix}{{\vartheta_{z}(y)} + {\vartheta_{z}\left( {y + L - {2b_{3}}} \right)} + {\vartheta_{z}\left( {y + L - {4b_{3}}} \right)}} \\{{+ \ldots} + {\vartheta_{z}\left\lbrack {y + L - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq {- {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2b_{3}}{L} \right)}} \right\rbrack}}}} & (39)\end{matrix}$

where fourth order of effects arising from the y dependence of Θ_(z)(y)have been neglected in Equations (37), (38), and (39) and Θ _(z)represents the average value of Θ_(z)(y) over the domain in y.

The lower and upper limits y₁ and y₂, respectively, of the domains in yfor SDP₁ ^(e), and SDP₂ ^(e), respectively, are

$\begin{matrix}{{y_{1} = {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)}}},} & (40) \\{y_{2} = {y_{0} - \frac{L}{2} + {2b_{2}}}} & (41)\end{matrix}$

where y₀ corresponds to the value of y at the middle of the domain in y.At the lower and upper limits y₁ and y₂, the respective SDP₁ ^(e) andSDP₂ ^(e) are expressed as

$\begin{matrix}{{{SDP}_{1}^{e}\left( {x,y_{1},\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y_{1}}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\pi\;\frac{y_{1}}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (42) \\{{x{\overset{\_}{\;\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y_{1},{\vartheta_{z} = 0}} \right)} \right\rbrack}} - {\left( {1 + {\eta\; q}} \right)E_{1}} + {\left( {1 + \eta} \right)E_{2}} +} & \; \\{{\eta\left( {q - 1} \right)E_{3}} + {\eta\left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right)} +} & \; \\{{\eta\; b_{3}\begin{Bmatrix}{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)}} \right\rbrack} + {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)} - {2b_{3}}} \right\rbrack}} \\{{+ \ldots} + {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}},} & \; \\{{{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y_{2}}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\pi\;\frac{y_{2}}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (43) \\{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y_{2},{\vartheta_{z} = 0}} \right)} \right\rbrack}} + {\left( {q - 1} \right)E_{1}} +} & \; \\{{\left( {1 + \eta} \right)E_{2}} - {\left( {q + \eta} \right)E_{3}} + \left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) +} & \; \\{{b_{3}\begin{Bmatrix}{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}}} \right\rbrack} + {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right\rbrack}} \\{+ {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right\rbrack}} \\{{+ \ldots} + {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}},} & \;\end{matrix}$

It is evident on examination of Equations (37), (42), and (43) that thecontribution of the surface figure terms to SDP and SDP₁ ^(e) arecontinuous at y₁ and that the contribution of the surface figure termsto SDP and SDP₂ ^(e) are continuous at y₂. This is a very significantproperty which is subsequently used in a procedure to eliminate theeffects of offset errors E₁, E₂, and E₃ and the effects of the ydependence of Θ_(z)(y).

As a consequence of the property, the ratio of the differences betweenSDP and SDP₁ ^(e) at y₁ and between SDP and SDP₂ ^(e) at y₂ isindependent of the surface figure except for the values of the surfacefigure error at the extreme edges of the domain being mapped.Expressions for the differences are obtained using Equations (37), (42),and (43) with the results

$\begin{matrix}{\left\lbrack {{{SDP}_{1}^{e}\left( {x,y_{1},\vartheta_{z}} \right)} - {{SDP}\left( y_{1} \right)}} \right\rbrack = {{{- \eta}\;{q\left( {E_{1} - E_{3}} \right)}} + {\eta\left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right)} + {\eta\;{b_{3}\begin{pmatrix}{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)} + {\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right)}} \\{{+ \ldots} + {\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2{qb}_{3}}} \right)}}\end{pmatrix}}}}} & (44) \\{\left\lbrack {{{SDP}\left( y_{2} \right)} - {{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)} -} \right\rbrack = {{- {q\left( {E_{1} - E_{3}} \right)}} + \left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) + {b_{3}{\quad\begin{pmatrix}{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}}} \right)} + {\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)}} \\\begin{matrix}{{+ {\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right)}} + \ldots +} \\{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}\end{matrix}\end{pmatrix}\quad}}}} & (45)\end{matrix}$

Using the relationship L=2qb₃ given by Equation (8), Equation (45) iswritten as

$\begin{matrix}{\left\lbrack {{{SDP}\left( y_{2} \right)} - {{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)} -} \right\rbrack = {{- {q\left( {E_{1} - E_{3}} \right)}} + \left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) + {{b_{3}\begin{pmatrix}{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)} + {\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right)}} \\{{+ \ldots} + {\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2{qb}_{3}}} \right\rbrack}}\end{pmatrix}}.}}} & (46)\end{matrix}$

The ratio of the value of the discontinuity D₁(y₀) given Equation (44)and the value of the discontinuity D₂(y₀) given by Equation (46) is thusequal to η, i.e.D₁(y₀)=ηD₂(y₀)  (47)whereD ₁(y ₀)≡[SDP ₁ ^(e)(x,y ₁,Θ_(z))−SDP(y ₁,Θ_(z))],  (48)D ₂(y ₀)≡[SDP(y ₂ ,Θ _(z))−SDP ₂ ^(e)(x,y ₂,Θ_(z))]  (49)

Based on the foregoing mathematical development, a general procedure fordetermining a surface figure function, ξ(y), without any prior knowledgeof the surface is 5 figure is as follows. first, select the length ofthe mirror that is to be mapped by selecting a value of q. The length ofthe mirror is given by Equation (8). Orient the stage mirror so thatθ_(z) is at or about zero and the distance x from the interferometer tothe stage mirror being mapped is relatively small. The small distancecan reduce the contribution to the geometric error correction term.While maintaining a nominal x distance to the stage mirror, acquiresimultaneous values for x₁, x₂, and x₃ while scanning the mirror in they-direction and monitor the position of the mirror in the y-directionand changes in stage orientation during the scan. The x₁, x₂, and x₃values, the position of the mirror in the y-direction, and the changesin stage orientation can be stored to the electronic controller's memoryor to disk. The stored data will be in the form of a 3×N array for thex₁, x₂, and x₃ information, where N is the total number of samplesacross the mirror, a 1×N array for y position information and a 1×Narray for changes in stage orientation information.

Next, calculate values for SDP, SDP₁ ^(e), and SDP₂ ^(e) on a samplinggrid (e.g., a uniform sampling grid) in preparation for a discreteFourier transform which will be performed later. These calculations caninclude averaging multiple values of SDP, SDP₁ ^(e), and SDP₂ ^(e) whereeach increment in the sampling grid corresponds to multiple values ofx₁, x₂, and x₃. Since the SDP₁ ^(e), and SDP₂ ^(e) parameters aresensitive to changes in stage orientation during the scan, the arraycontaining changes in stage orientation information during the scan isused to correct the SDP₁ ^(e) and SDP₂ ^(e) parameters to a chosenreference stage orientation.

Next, remove the discontinuities at the domain boundaries between SDP₁^(e) and SDP, and between SDP and SDP₂ ^(e) by adding appropriateconstant values to SDP₁ ^(e) and SDP₂ ^(e). These constant values are−D₁ and D₂ as given by Equations (48) and (49).

After the discontinuities have been removed, a resultant SDP array isconstructed by concatenating the SDP₂ ^(e) array with discontinuitysubtracted, SDP array and SDP₁ ^(e) array with discontinuity subtracted.The resultant SDP array spans the length of the mirror to be mapped, L.Next, perform a discrete Fourier transform on the resultant SDP array.Using a complex transfer function calculated for the interferometerbeing used, transform the SDP Fourier coefficients into the Fouriercoefficients for the surface figure function ξ(y). The surface figurefunction, ξ(y), can now be determined from the Fourier coefficientsusing Equation (13).

In certain embodiments, as discussed earlier, the interferometer axesspacing can be selected so that the double pass displacementmeasurements x₁, x₂, and x₃ measured by the interferometer have reducedor nominally zero sensitivity to spatial frequency componentscorresponding to zeros of the transfer function T(m). In embodimentswhere manufacturing tolerances or other constraints do not allow thisinterferometer axes spacing, the contributions of the surface figureerror function to the double pass distances x₁, x₂, and x₃ at thespatial frequencies for which the transfer function T(m) issignificantly reduced or zero are subsequently measured by alignmentwafer procedures that are based on measurements of properties of analignment wafer. The properties of transfer function T(m) are calculatedusing a measured value of η and yield in particular the values of thespatial frequencies for which the transfer function T(m) issignificantly reduced or zero.

The alignment wafer used in the alignment wafer procedures is generatedwith a linear array of alignment marks 62 such as shown diagrammaticallyin FIG. 4. The length of the linear array corresponds to the length ofthe scan in the y direction for which information about the surfacefigure is being determined and is represented as L_(a). In someembodiments, L_(a) is in a range from about 10 cm to about 100 cm (e.g.,about 20 cm or more, about 30 cm or more up to about 80 cm). The spacingof contiguous alignment marks is d_(a) which corresponds to a spatialfrequency 1/d_(a) that is 4 times a harmonic of a spatial frequency forwhich additional information about the surface figure is desired. Theharmonic may be the first harmonic of the spatial frequency if there areno significant effects of aliasing of higher harmonics in subsequentdata processing.

In some embodiments, the positions of the alignment marks arrangednominally parallel to the stage mirror are first measured in the xdirection by one or more of the three interferometers using an alignmentsensor to locate the alignment marks. Next, the linear array ofalignment marks is shifted relative to the stage mirror by d_(a) in adirection parallel to the stage mirror and the positions of thealignment marks are measured a second time by the one or more of thethree interferometers.

The selected value of the shift d_(a) corresponds to a π/2 phase shiftin the conjugated quadrature of a spatial frequency component of thesurface figure having a spatial wavelength of Λ=4d_(a). As a consequenceof the π/2 phase shift, the difference of the two measurements of thealignment mark positions in the x direction for each of the one or moreof the three interferometer axes contains to a high degree of accuracyonly the contribution of the spatial frequency components of the surfacefigure, i.e. the effects of departures of the positions of the array ofalignment marks from a datum line cancel out. In addition, the relativesensitivity of the difference of the two measurements to thecontribution of the spatial frequency components with spatial wavelengthΛ is of the order of ½.

The difference of the two measurements for each of the one or more ofthe three interferometer axes are analyzed for the Fourier seriescoefficients for the spatial frequency component having the spatialwavelength Λ. The resulting Fourier series coefficients correspond tothe respective spectral component of the surface figure with spatialwavelength Λ.

Without wishing to be bound by theory, a formal description of the firstalignment wafer procedure is next presented. The first array of measuredlocations of the linear array of alignment marks is represented byx_(i)(y_(j)) for i=1, 2, and/or 3 where y_(j) corresponds to thelocation in the y direction of alignment mark j where 1≦j≦N_(a). Thenumber of alignment marks, N_(a), is related to the scan length L_(a) inthe y direction as

$\begin{matrix}{N_{a} = {4{\frac{L_{a}}{\Lambda}.}}} & (50)\end{matrix}$

Note that N_(a) is an even integer mod 4. The second array of measuredlocations of the linear array of alignment marks is represented byx_(i)(y_(j)+d_(a)) for i=1, 2, and/or 3. The difference of the twoarrays may be expressed as

$\begin{matrix}{{\left\lbrack {{x_{i}\left( {y_{j} + d_{a}} \right)} - {x_{i}\left( y_{j} \right)}} \right\rbrack = {{\sum\limits_{m = 1}^{N}{{\overset{\sim}{A}}_{m}\left\{ {{\cos\left\lbrack {m\; 2\pi\frac{\left( {y_{j} + d_{a}} \right)}{L_{a}}} \right\rbrack} - {\cos\left( {m\; 2\;\pi\frac{y_{j}}{L_{a}}} \right)}} \right\}}} + {\sum\limits_{m = 1}^{N}{{\overset{\sim}{B}}_{m}\left\{ {{\sin\left\lbrack {m\; 2\pi\frac{\left( {y_{j} + d_{a}} \right)}{L_{a}}} \right\rbrack} - {\sin\left( {m\; 2\;\pi\;\frac{y_{j}}{L_{a}}} \right)}} \right\}}} + \overset{\_}{x} + {\Delta\;{\overset{\_}{\vartheta}}_{z}{d_{a}\left( {j - 1} \right)}}}},{1 \leq j \leq N_{a}},} & (51)\end{matrix}$where x is the shift in the x direction of alignment mark j=1 and Δ Θ_(z) is the change in the orientation of the alignment wafer,respectively, that are introduced when the alignment wafer is shifted inthe y direction by d_(a) and Ã_(m) and {tilde over (B)}_(m) are therespective Fourier coefficients corresponding to the specific doublepass measurement indicated by the index i. It is important to note thatthe Ã_(m) and {tilde over (B)}_(m) Fourier coefficients utilized hererepresent the contributions of the mirror surface figure error functionto the double pass displacement measurements and thus do not corresponddirectly to the Fourier coefficients of the surface figure errorfunction found in Equation (13). Equation (51) is rewritten usingtrigonometric identities as

$\begin{matrix}{{\left\lbrack {{x_{i}\left( {y_{j} + d_{a}} \right)} - {x_{i}\left( y_{j} \right)}} \right\rbrack = {{\sum\limits_{m = 1}^{N}{\begin{Bmatrix}{{{\overset{\sim}{A}}_{m}\left\lbrack {{\cos\left( {m\; 2\;\pi\frac{d_{a}}{L_{a}}} \right)} - 1} \right\rbrack} +} \\{{\overset{\sim}{B}}_{m}{\sin\left( {m\; 2\;\pi\frac{d_{a}}{L_{a}}} \right)}}\end{Bmatrix}{\cos\left\lbrack {m\; 2\;\pi\frac{y_{j}}{L_{a}}} \right\rbrack}}} + {\sum\limits_{m = 1}^{N}{\begin{Bmatrix}{{{- {\overset{\sim}{A}}_{m}}{\sin\left( {m\; 2\;\pi\frac{d_{a}}{L_{a}}} \right)}} +} \\{{\overset{\sim}{B}}_{m}\left\lbrack {{\cos\left( {m\; 2\;\pi\frac{d_{a}}{L_{a}}} \right)} - 1} \right\rbrack}\end{Bmatrix}{\sin\left( {m\; 2\;\pi\frac{y_{j}}{L_{a}}} \right)}}} + \overset{\_}{x} + {\Delta{\overset{\_}{\vartheta}}_{z}{d_{a}\left( {j - 1} \right)}}}},{1 \leq j \leq {N_{a}.}}} & (52)\end{matrix}$

Using the orthogonality properties of the trigonometric functions, thefollowing Fourier coefficients are obtained to a good approximation as

$\begin{matrix}{{\left( {{\overset{\sim}{A}}_{m^{\prime}} - {\overset{\sim}{B}}_{m^{\prime}}} \right) = {{- \left\{ {\frac{2}{N_{a}}{\sum\limits_{j = 1}^{N_{a}}{\left\lbrack {{x_{i}\left( {y_{j} + d_{a}} \right)} - {x_{i}\left( y_{j} \right)}} \right\rbrack{\cos\left\lbrack {m^{\prime}2\;\pi\frac{y_{j}}{L_{a}}} \right\rbrack}}}} \right\}}{\delta\left( {m^{\prime} - {N_{a}/4}} \right)}}},} & (53) \\{{\left( {{\overset{\sim}{A}}_{m^{\prime}} + {\overset{\sim}{B}}_{m^{\prime}}} \right) = {{{- \left\{ {\frac{2}{N_{a}}{\sum\limits_{j = 1}^{N_{a}}{\left\lbrack {{x_{i}\left( {y_{j} + d_{a}} \right)} - {x_{i}\left( y_{j} \right)}} \right\rbrack{\sin\left\lbrack {m^{\prime}2\;\pi\frac{y_{j}}{L_{a}}} \right\rbrack}}}} \right\}}{\delta\left( {m^{\prime} - {N_{a}/4}} \right)}} - {\Delta{\overset{\_}{\vartheta}}_{z}d_{a}}}},} & (54)\end{matrix}$where δ(m) is the Kronecker delta function, i.e. δ(m)=1, m=0 and δ(m)=0,m≠0. Ã_(m′) and {tilde over (B)}_(m′) as indicated here correspond tothe specific double pass measurement indicated by the index i, and thusvalues of Ã_(m′) and {tilde over (B)}_(m′) must be calculated separatelyfor each of the double pass measurements for which a surface figureerror correction corresponding to the spatial frequency

$\frac{N_{a}}{4L_{a}}$is desired. The value of the constant term Δ Θ _(z)d_(a) in Equation(54) is determined from the product of Δ Θ _(z) obtained from a linearfit to the difference [x_(i)(y_(j)+d_(a))−x_(i)(y_(j))] and d_(a)obtained from the measured positions of the alignment marks in the ydirection. With the determined value of Δ Θ _(z)d_(a), values for Ã_(m′)and {tilde over (B)}_(m′) for m′=N_(a)/4 are obtained from Equations(53) and (54).

The coefficients Ã_(m′) and {tilde over (B)}_(m′) as obtained can thenbe used to generate corrections for the corresponding double passmeasurement for those components of the surface figure error functionhaving spatial frequency

$\frac{N_{a}}{4L_{a}}.$

The effects of aliasing have been omitted, for example, in Equations(53) and (54). In embodiments, the effects of aliasing can be evaluatedin an end use application by repeating the steps of the alignment waferprocedure with an alignment wafer that has a d_(a) corresponding to ¼ ofthe spatial wavelength of the spectral component responsible for theundesired aliased component.

In certain embodiments, the positions of the alignment marks arrangednominally parallel to the stage mirror are first measured in the xdirection by one or more of the three interferometers. Next, thealignment wafer is rotated by 180 degrees and the positions of thealignment marks are measured a second time.

For the analysis in such embodiments, the measured position of analignment mark from the first measurement is added with the measuredposition of a conjugate alignment mark from the second measurementwherein an alignment mark and its conjugate alignment mark have the samenominal position in the y direction. The analysis for data acquired insuch procedures is the same as the analysis described above.

In some embodiments, this alignment wafer procedure can be used todetermine surface figures for mirrors in a lithography tool at timeswhen the tool is not being used to expose wafers. For example, thisprocedure can be used during routine maintenance of the tool. At suchtimes, a dedicated scan sequence can be implemented, allowing a portionof a mirror to be scanned at a rate optimized for mirrorcharacterization purposes. Moreover, multiple scans can be performed fordifferent relative distances between the mirror and the interferometerand at different nominal stage orientations, providing a family ofultimate surface figures which can be interpolated for use at otherrelative distances between the mirror and the interferometer and atother stage orientations.

In embodiments where data is acquired during dedicated scan sequences,the data can be acquired over time periods that are relatively long withrespect to various sources of random errors in x₁, x₂, and x₃measurements. Accordingly, uncertainty in surface figures due to randomerrors sources can be reduced. For example, one source of random errorsare fluctuations in the composition or density of the atmosphere in theinterferometer beam paths. Over relatively long time periods, thesefluctuations average to zero. Accordingly, acquiring data forcalculating SDP and SDP^(e) values over sufficiently long periods cansubstantially reduce the uncertainty of ξ(y) due to these non-stationaryatmospheric fluctuations.

In certain embodiments where the mirror is to be mapped in situ during,for example, lithography tool operation, arrays of x₁, x₂, and x₃, yposition and stage orientation can be acquired and stored to theelectronic controller's memory or to disk for specific relativedistances and nominal stage orientations. After the arrays of data havebeen acquired, the remaining steps for mapping the surface figure arethe same as in an offline analysis. Values for SDP, SDP₁ ^(e), and SDP₂^(e) can be calculated and assigned to a sampling grid (e.g., a uniformsampling grid) in preparation for a discrete Fourier transform whichwill be performed later. These calculations can include averagingmultiple values of SDP, SDP₁ ^(e), and SDP₂ ^(e) where each increment inthe sampling grid corresponds to multiple values of x₁, x₂, and x₃.Since the SDP₁ ^(e), and SDP₂ ^(e) parameters are sensitive to changesin stage orientation during the scan, the array containing changes instage orientation information during the scan is used to correct theSDP₁ ^(e) and SDP₂ ^(e) parameters to a chosen reference stageorientation.

Next, the discontinuities at the domain boundaries between SDP₁ ^(e) andSDP, and between SDP and SDP₂ ^(e) are removed by adding appropriateconstant values to SDP₁ ^(e) and SDP₂ ^(e). These constant values are−D₁ and D₂ as given by Equations (48) and (49).

After the discontinuities have been removed, a resultant SDP array isconstructed by appropriately concatenating the SDP₂ ^(e), SDP and SDP₁^(e) arrays. The resultant SDP array spans the length of the mirror tobe mapped, L. Next, perform a discrete Fourier transform on theresultant SDP array. Using a complex transfer function calculated forthe interferometer being used, transform the SDP Fourier coefficientsinto the Fourier coefficients for the surface figure function ξ(y). Thesurface figure function, ξ(y), can now be determined from the Fouriercoefficients using Equation (13).

The surface figure function may be updated and maintained by computing arunning average of the surface figure function acquired at successivetime intervals. The running average can be stored to the electroniccontroller's memory or to disk. The optimum time interval which a givenaverage spans can be chosen after determining the time scales over whichthe surface figure function ξ(y) significantly changes.

In certain other embodiments, a predetermined surface figure function ξcan be updated or variations in ξ can be monitored based on values ofSDP and/or SDP^(e)s calculated from newly-acquired interferometry data.For example, in a lithography tool, x₁, x₂, and x₃ values acquiredduring operation of the tool (e.g., during wafer exposure) can be usedto update a mirror's surface figure, or can be used to monitor changesin the surface figure and provide an indicator of when new full mirrorcharacterizations are required to maintain the interferometry system'saccuracy.

In implementations where the surface figure function is updated duringan exposure sequence of a tool, data can be acquired over multiple waferexposure sequences so that changes that occur to the surface figurefunction as a result of systematic changes in the tool that occur eachsequence can be distinguished from actual permanent changes in themirror's characteristics. One example of systematic changes in a toolthat may contribute to variations in a surface figure function arestationary changes in the atmosphere in the interferometer's beam paths.Such changes occur for each wafer exposure cycle as the compositionand/or gas pressure inside the lithography tool is changed. In theabsence of any other technique for monitoring and/or compensating forthese effects, they can result in errors in the x₁, x₂, and x₃measurements that ultimately manifest as errors in the surface figurefunction. However, since the changes to the atmosphere are stationary,their effect on ξ(y) can be reduced (e.g., eliminated) by identifyingvariations that occur to ξ(y) with the same period as the wafer exposurecycle, and subtracting these variations from the correction that isultimately made to a stage measurement.

The predetermined surface figure function in the certain otherembodiments may be updated by creating a running average of the surfacefigure function using the predetermined surface figure function andsurface figure functions acquired in situ at subsequent time intervals.The subsequent surface figure functions acquired in situ may be acquiredas in the certain embodiments description above. The running average canbe stored to the electronic controller's memory or to disk. The optimumtime interval which a given average spans can be chosen afterdetermining the time scales over which the surface figure function ξ(y)significantly changes.

An alternative procedure for updating a predetermined surface figure inthe certain other embodiments above may be used. The first step in thealternate procedure is to measure values of SDP(y, Θ _(z)=0) for asection of the x-axis stage mirror. The length of the section or domainis ≧4b₃ and can include the entire length of the stage mirror. Next,integrals of SDP(y, Θ _(z)=0) with weight functions cos(s2πy/L) and sin(s2πy/L) are computed from the measured values of SDP(y, Θ _(z)=0) minusthe mean value <SDP(y, Θ _(z)=0)> of SDP (y, Θ _(z)=0), SDP₁ ^(e)(y, Θ_(z)=0)−ηD(y₀), and SDP₂ ^(e)(y, Θ _(z)=0)+D(y₀), <SDP(y, Θ _(z)=0)≦.The integrals are:

$\begin{matrix}{A_{s}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}{\quad{{\left\lbrack {{{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z} = 0}} \right)} - \left\langle {{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z} = 0}} \right)} \right\rangle} \right\rbrack{\sin\left( \frac{s\; 2\pi\; y}{L} \right)}\ {\mathbb{d}y}},{s \geq 1},}}}}} & (52) \\{B_{s}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}{\quad{{\left\lbrack {{{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z} = 0}} \right)} - \left\langle {{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z} = 0}} \right)} \right\rangle} \right\rbrack{\sin\left( \frac{s\; 2\pi\; y}{L} \right)}\ {\mathbb{d}y}},{s \geq 1.}}}}}} & (53)\end{matrix}$

The integrals expressed by Equations (52) and (53) are evaluated usingthe series representation of SDP given by Equation (16) with the results

$\begin{matrix}{{A_{q}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}{\begin{bmatrix}{{\sum\limits_{m = 1}^{N}{A_{m}^{\prime}{\cos\left( \frac{m\; 2\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\pi\; y}{L} \right)}}} +} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}{\sin\left( \frac{m\; 2\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\pi\; y}{L} \right)}}}\end{bmatrix}\ {\mathbb{d}y}}}}},{q \geq 1},} & (54) \\{B_{q}^{''} = {{\frac{4}{\left( {L - {2\; b_{3}}} \right)} \times {\int_{{({{- \frac{L}{2}}\; + {2\; b_{2}}})}\;}^{\lbrack{\frac{L}{2} - {2\;{({b_{3} - b_{2}})}}}\rbrack}{\begin{bmatrix}{{\sum\limits_{m = 1}^{N}{{\; A_{m}^{\prime}}\;{\cos\left( \frac{m\; 2\;\pi\; y}{L} \right)}\;{\sin\left( \frac{q\; 2\;\pi\; y}{L} \right)}}} +} \\{\sum\limits_{m = 1}^{N}{{\; B_{m}^{\prime}}\;{\sin\left( \frac{m\; 2\;\pi\; y}{L} \right)}\;{\sin\left( \frac{q\; 2\;\pi\; y}{L} \right)}}}\end{bmatrix}\ {\mathbb{d}y}\mspace{14mu} q}}} \geq 1.}} & (55)\end{matrix}$

where coefficients A′_(q) and B′_(q) are with respect to SDP(y, Θ_(z)=0) with <SDP(y, Θ _(z)=0)> subtracted. The evaluation of theintegrals in Equation (54) is next performed with the results

$\begin{matrix}{{A_{q}^{''} = {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times \begin{Bmatrix}{{\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{1}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} +} \\{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} +} \\{\frac{1}{\left( {q - m} \right)2\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} +} \\{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}} -} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{\frac{1}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} -} \\{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} -} \\{\frac{1}{\left( {q - m} \right)2\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} -} \\{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}\end{Bmatrix}}},{q \geq 1},} & (56) \\{{{A_{q}^{''} = {A_{q}^{\prime} + {\frac{1}{\left( {1\; - \frac{2\; b_{3}}{L}} \right)} \times \begin{Bmatrix}{{\underset{m \neq q}{\sum\limits_{{m = 1},}^{N}}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} +} \\{\sin\;{\pi\left( {q + m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} +} \\{{\sin\;{\pi\left( {q - m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\;}\end{Bmatrix}}\end{pmatrix} \right)}} -} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q - m}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} -} \\{{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {4\;\frac{b_{2}}{L}} \right\rbrack}}\;}\end{Bmatrix}} -} \\{\frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack \;{4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} -} \\{\cos\;{\pi\left( {q - m} \right)}\left( {4\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}\end{Bmatrix}}}},{q \geq 1},}\;} & (57) \\{{{A_{q}^{''} = {A_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2\; b_{3}}{L}} \right)} \times \left\{ \begin{matrix}{{\underset{m \neq q}{\sum\limits_{{m = 1},}^{N}}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\;\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} +} \\{\sin\;\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} +} \\{\sin\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}} -} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q - m}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} -} \\{\cos\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} -} \\{\frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} -} \\{\cos\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}}},{q \geq 1},}\mspace{56mu}} & (58) \\{{{A_{q}^{''} = {A_{q}^{\prime} + {\frac{\left( \frac{2b_{3}}{L} \right)}{\left\lbrack {1 - \left( \frac{2\; b_{3}}{L} \right)} \right\rbrack} \times \left\{ \begin{matrix}{{\underset{m \neq q}{\sum\limits_{{m = 1},}^{N}}{A_{m}^{\prime}\left( \begin{pmatrix}{{\left( {- 1} \right)^{q + m + 1}\begin{Bmatrix}{\sin\;{c\;\left\lbrack {{\pi\left( {q + m} \right)}\;\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\cos\left\lbrack {{\pi\left( {q + m} \right)}\;\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\cos\left\lbrack \;{{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}} +} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q + m} \right)}\;\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\sin\left\lbrack {{\pi\left( {q + m} \right)}\;\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} -} \\{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\sin\left\lbrack {{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}}},{q \geq 1.}}\;} & (59)\end{matrix}$

The evaluation of the integrals in Equation (55) is next performed withthe results

$\begin{matrix}{{{B_{q}^{''} = {\frac{1}{\left( {1 - \frac{2\; b_{3}}{L}} \right)} \times \left\{ \begin{matrix}{{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{1}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q + m} \right)}\;\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} -} \\{\cos\;{{\pi\left( {q + m} \right)}\;\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} +} \\{\frac{1}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} -} \\{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}} +} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{{- \frac{1}{\left( {q + m} \right)\; 2\;\pi}}\begin{Bmatrix}{{\sin\;{{\pi\left( {q + m} \right)}\;\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} +} \\{\sin\;{{\pi\left( {q + m} \right)}\;\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} +} \\{\frac{1}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} +} \\{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}},{q \geq 1},}\mspace{45mu}} & (60) \\{{{B_{q}^{''} = {B_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2\; b_{3}}{L}} \right)} \times \left\{ \begin{matrix}{{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q + m}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q + m} \right)}\;\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} -} \\{\cos\;{{\pi\left( {q + m} \right)}\;\left\lbrack {4\frac{b_{2}}{L}} \right\rbrack}}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} -} \\{\cos\;{\pi\left( {q - m} \right)}\left( {4\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}} +} \\{\underset{m \neq q}{\sum\limits_{m = 1}^{N}}{B_{m}^{\prime}\left( \begin{pmatrix}{{{- \frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)\; 2\;\pi}}\begin{Bmatrix}{{\sin\;{{\pi\left( {q + m} \right)}\;\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} +} \\{\sin\;{\pi\left( {q + m} \right)}\;\left( {4\frac{b_{2}}{L}} \right)}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} +} \\{\sin\;{\pi\left( {q - m} \right)}\left( {4\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}}},{q \geq 1},}\mspace{56mu}} & (61) \\{{{B_{q}^{''} = {B_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2\; b_{3}}{L}} \right)} \times \left\{ \begin{matrix}{{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{{\frac{\left( {- 1} \right)^{q + m}}{\left( {q + m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} -} \\{\cos\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\cos\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} -} \\{\cos\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}}} +} \\{\underset{m \neq q}{\sum\limits_{{m = 1},}^{N}}{B_{m}^{\prime}\left( \begin{pmatrix}{{{- \frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)\; 2\;\pi}}\begin{Bmatrix}{{\sin\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} +} \\{\sin\left\lbrack {2{\pi\left( {q + m} \right)}\;\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)\; 2\;\pi}\begin{Bmatrix}{{\sin\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} +} \\{\sin\left\lbrack {2\;{\pi\left( {q - m} \right)}\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}}},{q \geq 1.}}\mspace{59mu}} & (62) \\{{{B_{q}^{''} = {B_{q}^{\prime} + {\frac{\left( \frac{2b_{3}}{L} \right)}{\left\lbrack {1 - \left( \frac{2\; b_{3}}{L} \right)} \right\rbrack} \times \left\{ \begin{matrix}{{\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{{\left( {- 1} \right)^{q + m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q + m} \right)}\;\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\sin\left\lbrack {{\pi\left( {q + m} \right)}\;\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\sin\left\lbrack {{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}} +} \\{\underset{m \neq q}{\sum\limits_{{m = 1},}^{N}}{B_{m}^{\prime}\left( \begin{pmatrix}{{{- \left( {- 1} \right)^{q + m + 1}}\begin{Bmatrix}{\sin\;{c\left\lbrack {{\pi\left( {q + m} \right)}\;\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\cos\left\lbrack {{\pi\left( {q + m} \right)}\;\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}} +} \\{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{\sin\;{c\left\lbrack \;{{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times} \\{\cos\left\lbrack {{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}\end{Bmatrix}}\end{pmatrix} \right)}}\end{matrix} \right\}}}},{q \geq 1.}}\;} & (63)\end{matrix}$

The effects of non-orthogonality of functions cos(m2πy/L) andsin(m2πy/L) over the domain of y in SPD (y, Θ _(z)=0) are evident uponexamination of Equations (59) and (63). The effects of non-orthogonalityare represented by the differences (A″_(q)−A′_(q)) and (B″_(q)−B′_(q))in Equations (59) and (63), respectively.

Equations (59) and (63) can be solved for the A′_(q) and B′_(q),respectively, by an iterative procedure. The degree to which the effectsof non-orthogonality are decoupled from the leading A′_(q) and B′_(q)terms in Equations (59) and (63) depends on the ratio (2b₃/L). Thelarger coefficients of “off diagonal terms” in Equations (59) and (63)generally occur for small values |q−m| and the relative magnitude r ofthe coefficients of these terms is

$\begin{matrix}{r \cong {\frac{\left( \frac{2b_{3}}{L} \right)}{1 - \left( \frac{2b_{3}}{L} \right)}\sin\;{{c\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}.}}} & (64)\end{matrix}$

The iterative procedure is a nontrivial step for which two solutions aregiven. The step is nontrivial since cos(m2πy/L) and sin (m2πy/L) arefunctions that are not orthogonal as a set over the domain ofintegration in y. One solution involves using a surface figure functionξ obtained at an earlier time based on measured values of SDP, SDP₁^(e), and SDP₂ ^(e) to compute values for the off-diagonal terms inEquations (59) and (63).

Alternatively, or additionally, information about a surface figure canbe obtained using other techniques, such as based on values of a FDPsuch as given by Equation (3) or from interferometric measurements madeex-situ, e.g., a Fizeau interferometer, before the stage mirror isinstalled in a lithography tool.

The resulting values for A′_(m) and B′_(m) may be used without anyiteration if the resulting values for A′_(m) and B′_(m) indicate thatthere has not been any significant change in the surface figure functionξ. If there has been a significant change, the resulting values forA′_(m) and B′_(m) are used as input information to compute theoff-diagonal terms in a second step of the iterative procedure. Theiterative steps are repeated until stable, i.e., asymptotic, values forA′_(m) and B′_(m) are obtained. The values for A_(m) and B_(m) aresubsequently obtained using Equations (17) and (18) from thecoefficients A′_(m) and B′_(m) produced by the iterative procedure.

After the surface figure function ξ(y, Θ_(z)=0) is determined for thespecified section of the x-axis stage mirror, the procedure can berepeated for other sections of the x-axis stage mirror that are used inthe respective lithography tool.

In some embodiments, measured values of SDP can be used to extend aknown surface figure function ξ to other sections of the x-axis stagemirror. Equation (7) allows the generation of figure error function ξ(y,Θ_(z)=0) for the other sections of the x-axis stage mirror from theknown surface figure function ξ(y, Θ_(z)=0) in an adjacent section ofthe x-axis stage mirror.

In certain embodiments, local spatial derivatives of a surface figurefunction can be measured using the methods disclosed herein, in somecases, with relatively high sensitivity. The sensitivity of the thirdorder term represented by the last term in Equation (16) is proportionalto m and is equal, for example, to the sensitivity of the remainingfirst order terms in Equation (16) for m≈100 for |Θ_(z)|=0.5milliradians and x=0.7. To measure the local spatial derivativesdirectly or with a higher sensitivity, the procedure described for thedetermination of ξ(y, Θ_(z)=0) can be repeated for non-zero values ofΘ_(z) and for large values of x and y for the x-axis and y-axis stagemirrors, respectively.

The surface figure function ξ(y, Θ _(z)=0) obtained in the abovedescribed procedures includes in particular information about ξ(y, Θ_(z)=0) that is quadratic in y. However, the surface figure functionξ(y, Θ _(z)=0) obtained in the above described procedure does notrepresent any error in ξ(y, Θ _(z)=0) that is linear in y since SDP iszero for a plane mirror surface.

The error in the determined ξ(y, Θ _(z)=0) linear in y relative to theerror in the determined ξ(x, Θ _(z)=0) linear in x for the x-axis andthe y-axis stage mirrors, respectively, i.e., a differential error, isthe only portion of the linear error that is relevant. A common modeerror corresponds to a rotation of the stage. The differential error isrelated to the angle between the x-axis and y-axis stage mirrors and theeffect of the differential error is compensated by measuring the anglebetween the two stage mirrors. The angle between the two stage mirrorscan be measured by use of an alignment wafer and the alignment waferrotated by 90 degrees for each plane defined by the multiple-axes/planeinterferometers.

It is the information about the angles between the two stage mirrorsthat is used for compensating surface figure functions in yaw.Additional information about the respective surface figure functions mayalso be obtained by introducing a displacement of the second passmeasurement beams at the stage mirror by changing the orientation of thestage mirror to change the respective pitch and repeating the proceduresdescribed herein to determine the surface figure functions at theposition of the sheared measurement beams.

Values of Θ_(z)(y) are measured by the y-axis interferometer. During ascan of the x-axis mirror, typically there is nominally no change in thelocation stage in the x direction. Accordingly, there is no change inthe nominal location of the measurement beams of the y-axisinterferometer on the surface of the corresponding y-axis stage mirror.However, the relative scales of the measurement axes of the y-axisinterferometer used to measure Θ_(z)(y) will not, in general, beidentical and furthermore may be a function of y. The respective scalesmay be different due to geometric effects such as described in commonlyowned U.S. patent application Ser. No. 10/872,304 entitled “COMPENSATIONFOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRRORINTERFEROMETER METROLOGY SYSTEMS” or due to stationary systematicchanges in the density gradients of the gas in the y-axis interferometermeasurement beam paths.

Such errors due to relative scale errors can be difficult to measure andare typically done using an alignment wafer. However, measured values ofthe SDP and values of SDP reconstructed from the derived mirror surfacefigure functions can be used to test to see if such errors are presentand if so, measure the y dependence of the relative scale errors thatare quadratic and higher order in terms of y. Over time scales shortcompared to the time scales for changes in (E₁-E₃), the differencebetween measured values of SDP and values of SDP reconstructed from thederived mirror surface figure functions are obtained as a function of y.Relative scaling errors in the y axes interferometer measurements whichresult in Θ_(z)(y) having a quadratic error term with respect to y canbe identified as a term linear in y for the difference between measuredand reconstructed values of SDP. Similarly errors in Θ_(z)(y) which arecubic and higher order with respect to y can be identified as quadraticand higher order terms in y for the difference between measured andreconstructed values of SDP.

The procedure described for determining and compensating errors in themeasured values of Θ_(z)(y) can also include compensating the effects ofstationary effects of the gas in the respective measurement andreference beam paths, such as described in U.S. patent application Ser.No. 11/365,991, entitled “INTERFEROMETRY SYSTEMS AND METHODS OF USINGINTERFEROMETRY SYSTEMS,” filed on Mar. 1, 2006, the entire contents ofwhich is incorporated herein by reference.

The surface figure function for an x-axis stage mirror will, in general,be a function of both y and z and the surface figure function for they-axis stage mirror will in general be a function of both x and z. Thegeneralization to cover the z dependent properties is subsequentlyaddressed by the use of a two three-axis/plane interferometer system inconjunction with a procedure to measure the angle between the x-axis andy-axis stage mirrors in two different parallel planes corresponding tothe two planes defined by the two three-axis/plane interferometersystem.

Once determined, surface figure functions are typically used tocompensate interferometry system measurements, thereby improving aninterferometry system's accuracy. For example, ξ(y) can be used tocompensate measurements of the position of an x-axis stage mirror along,e.g., axis x₁ for a position y based on the following equations. If d isthe true x-axis displacement of the stage mirror and ξ′(y) is the truesurface figure function, an uncorrected measurement at position y, willgive, for example,

$\begin{matrix}{{{{\overset{\sim}{x}}_{1}(y)} = {d + \left( \frac{{\xi_{1}^{\prime}(y)} + {\xi_{0}^{\prime}(y)}}{2} \right) + {d_{nom}{\theta_{z}\left( {\frac{\partial\xi_{1}^{\prime}}{\partial y} - \frac{\partial\xi_{0}^{\prime}}{\partial y}} \right)}}}},} & (65)\end{matrix}$where the lowest order terms dependent on the mirror surface figurefunction have been shown. ξ′₁ and ξ′₀ refer to the surface figurefunction values at positions x′₁ and x′₀, respectively, and d_(nom) is anominal displacement value used to calculate the geometric error termthat occurs for non-zero θ_(z).

Using the surface figure function, a compensated displacement x_(c)(y)can be calculated from the measured displacement based on the followingequation:

$\begin{matrix}{{x_{c}(y)} = {{{\overset{\sim}{x}}_{1}(y)} - \left( \frac{{\xi_{1}(y)} + {\xi_{0}(y)}}{2} \right) - {d_{nom}{{\theta_{z}\left( {\frac{\partial\xi_{1}}{\partial y} - \frac{\partial\xi_{0}}{\partial y}} \right)}.}}}} & (66)\end{matrix}$where the lowest order terms dependent on the mirror surface figure havebeen shown. In embodiments, various other error compensation techniquescan be used to reduce other sources of error in interferometermeasurements. For example, cyclic errors that are present in the lineardisplacement measurements can be reduced (e.g., eliminated) and/orcompensated by use of one of more techniques such as described incommonly owned U.S. patent application Ser. No. 10/097,365, entitled“CYCLIC ERROR REDUCTION IN AVERAGE INTERFEROMETRIC MEASUREMENTS,” andU.S. patent application Ser. No. 10/616,504 entitled “CYCLIC ERRORCOMPENSATION IN INTERFEROMETRY SYSTEMS,” which claims priority toProvisional Patent Application No. 60/394,418 entitled “ELECTRONICCYCLIC ERROR COMPENSATION FOR LOW SLEW RATES,” all of which are by HenryA. Hill and the contents of which are incorporated herein in theirentirety by reference.

An example of another cyclic error compensation technique is describedin commonly owned U.S. patent application Ser. No. 10/287,898 entitled“INTERFEROMETRIC CYCLIC ERROR COMPENSATION,” which claims priority toProvisional Patent Application No. 60/337,478 entitled “CYCLIC ERRORCOMPENSATION AND RESOLUTION ENHANCEMENT,” by Henry A. Hill, the contentsof which are incorporated herein in their entirety by reference.

Another example of a cyclic error compensation technique is described inU.S. patent application Ser. No. 10/174,149 entitled “INTERFEROMETRYSYSTEM AND METHOD EMPLOYING AN ANGULAR DIFFERENCE IN PROPAGATION BETWEENORTHOGONALLY POLARIZED INPUT BEAM COMPONENTS,” which claims priority toProvisional Patent Application 60/303,299 entitled “INTERFEROMETRYSYSTEM AND METHOD EMPLOYING AN ANGULAR DIFFERENCE IN PROPAGATION BETWEENORTHOGONALLY POLARIZED INPUT BEAM COMPONENTS,” both by Henry A. Hill andPeter de Groot, the contents both of which are incorporated herein intheir entirety by reference.

A further example of a cyclic error compensation technique is describedin commonly owned Provisional Patent Application No. 60/314,490 filedentitled “TILTED INTERFEROMETER,” by Henry A. Hill, the contents ofwhich is herein incorporated in their entirety by reference.

Other techniques for cyclic error compensation include those describedin U.S. Pat. No. 6,137,574 entitled “SYSTEMS AND METHODS FORCHARACTERIZING AND CORRECTING CYCLIC ERRORS IN DISTANCE MEASURING ANDDISPERSION INTERFEROMETRY;” U.S. Pat. No. 6,252,668 B1, entitled“SYSTEMS AND METHODS FOR QUANTIFYING NON-LINEARITIES IN INTERFEROMETRYSYSTEMS;” and U.S. Pat. No. 6,246,481, entitled “SYSTEMS AND METHODS FORQUANTIFYING NONLINEARITIES IN INTERFEROMETRY SYSTEMS,” wherein all threeare by Henry A. Hill, the contents of the three above-cited patents andpatent applications are herein incorporated in their entirety byreference.

Improved statistical accuracy in measured values of SDP can be obtainedby taking advantage of the relatively low bandwidth of measured valuesof SDP compared to the bandwidth of the corresponding lineardisplacement measurements using averaging or low pass filtering. Therelatively low bandwidth arises because of SDP invariance with respectto displacements of the mirror along the measurement axes and itinvariance (at least to second order) on rotations of the mirror.Variations in SDP occur primarily as a result of variations of themirror surface figure as the mirror is scanned orthogonally to themeasurement axes, which depends on the mirror scan speed, but generallyoccurs at much lower rates than the sampling rates of the detectors usedin the three-axis/plane interferometer. For example, where the 1/e² beamdiameter is 5 mm and the stage is scanned at a rate of 0.5 m/s, thebandwidth of measured values of SDP will be of the order of 100 Hzcompared to typical sampling rates of the respective detectors of theorder of 10 MHz.

The effects of offset errors in the measured values of SDP can bemeasured by use of an alignment wafer and the alignment wafer rotated by180 degrees and the offset error effects compensated. Offset errorsarise in SDP because SDP is derived from three different interferometermeasurements where each of the three interferometers can only measurerelative changes in a respective reference and measurement beam paths.In addition, the offset errors may change with time because thecalibrations of each of the three interferometers may change withrespect to each other, e.g. due to changes in temperature. An alignmentwafer is a wafer that includes several alignment marks that areprecisely spaced with respect to each other. Accordingly, a metrologysystem can be calibrated by locating the marks with the system'salignment sensor and comparing the displacement between each alignmentmark as measured using the metrology system with the known displacement.Because of offset errors in the interferometer distance measurements andbecause the angle between the measurement axes of the x-axis and y-axisinterferometers is not generally known, the angle between the x-axis andy-axis stage mirrors should be independently measured. This angle can bemeasured in one or more planes according to whether interferometersystem 10 comprises one or two three-axis/plane interferometers by useof an alignment wafer and the alignment wafer rotated by 90 degrees.

While the foregoing description is with regard to a particularinterferometer assembly, namely interferometer 100, in general, otherassemblies can also be used to obtain values for SDP and otherparameters. For example, in some embodiments, an interferometer assemblycan be configured to monitor the position of a measurement object alongmore than three coplanar axes (e.g., four or more axes, five or moreaxes). Moreover, while interferometer includes non-coplanar measurementaxes, other embodiments can include exclusively coplanar axes.Furthermore, the relative position of the common measurement beam pathis not limited to the position in interferometer 100. For example, insome embodiments, the common measurement beam path can be an outermostpath, instead of being flanked by beam paths on either side within thecommon plane.

In certain embodiments, individual, rather than compound, opticalcomponents can be used. For example, free-standing beam splitters can beused to divide the first path measurement beam into the othermeasurement beams. Such a configuration may allow one to adjust therelative spacing of the beams, and hence the relative spacing of themeasurement axes in a multi-axis interferometer.

In some embodiments, multiple single axis interferometers can be usedinstead of a multi-axis interferometer. For example, a three-axis/planeinterferometer can be replaced by three single axis interferometers(e.g., high stability plane mirror interferometers), arranged so thateach interferometer's axis lies in a common plane.

As discussed previously, lithography tools are especially useful inlithography applications used in fabricating large scale integratedcircuits such as computer chips and the like. Lithography is the keytechnology driver for the semiconductor manufacturing industry. Overlayimprovement is one of the five most difficult challenges down to andbelow 100 nm line widths (design rules), see, for example, theSemiconductor Industry Roadmap, p. 82 (1997).

Overlay depends directly on the performance, i.e., accuracy andprecision, of the distance measuring interferometers used to positionthe wafer and reticle (or mask) stages. Since a lithography tool mayproduce $50-100M/year of product, the economic value from improvedperformance distance measuring interferometers is substantial. Each 1%increase in yield of the lithography tool results in approximately$1M/year economic benefit to the integrated circuit manufacturer andsubstantial competitive advantage to the lithography tool vendor.

The function of a lithography tool is to direct spatially patternedradiation onto a photoresist-coated wafer. The process involvesdetermining which location of the wafer is to receive the radiation(alignment) and applying the radiation to the photoresist at thatlocation (exposure).

To properly position the wafer, the wafer includes alignment marks onthe wafer that can be measured by dedicated sensors. The measuredpositions of the alignment marks define the location of the wafer withinthe tool. This information, along with a specification of the desiredpatterning of the wafer surface, guides the alignment of the waferrelative to the spatially patterned radiation. Based on suchinformation, a translatable stage supporting the photoresist-coatedwafer moves the wafer such that the radiation will expose the correctlocation of the wafer.

During exposure, a radiation source illuminates a patterned reticle,which scatters the radiation to produce the spatially patternedradiation. The reticle is also referred to as a mask, and these termsare used interchangeably below. In the case of reduction lithography, areduction lens collects the scattered radiation and forms a reducedimage of the reticle pattern. Alternatively, in the case of proximityprinting, the scattered radiation propagates a small distance (typicallyon the order of microns) before contacting the wafer to produce a 1:1image of the reticle pattern. The radiation initiates photo-chemicalprocesses in the resist that convert the radiation pattern into a latentimage within the resist.

Interferometry metrology systems, such as those discussed previously,are important components of the positioning mechanisms that control theposition of the wafer and reticle, and register the reticle image on thewafer. If such interferometry systems include the features describedabove, the accuracy of distances measured by the systems can beincreased and/or maintained over longer periods without offlinemaintenance, resulting in higher throughput due to increased yields andless tool downtime.

In general, the lithography system, also referred to as an exposuresystem, typically includes an illumination system and a waferpositioning system. The illumination system includes a radiation sourcefor providing radiation such as ultraviolet, visible, x-ray, electron,or ion radiation, and a reticle or mask for imparting the pattern to theradiation, thereby generating the spatially patterned radiation. Inaddition, for the case of reduction lithography, the illumination systemcan include a lens assembly for imaging the spatially patternedradiation onto the wafer. The imaged radiation exposes resist coatedonto the wafer. The illumination system also includes a mask stage forsupporting the mask and a positioning system for adjusting the positionof the mask stage relative to the radiation directed through the mask.The wafer positioning system includes a wafer stage for supporting thewafer and a positioning system for adjusting the position of the waferstage relative to the imaged radiation. Fabrication of integratedcircuits can include multiple exposing steps. For a general reference onlithography, see, for example, J. R. Sheats and B. W. Smith, inMicrolithography: Science and Technology (Marcel Dekker, Inc., New York,1998), the contents of which is incorporated herein by reference.

Interferometry systems that utilize the mirror mapping proceduresdescribed above can be used to precisely measure the positions of eachof the wafer stage and mask stage relative to other components of theexposure system, such as the lens assembly, radiation source, or supportstructure. In such cases, the interferometry system can be attached to astationary structure and the measurement object attached to a movableelement such as one of the mask and wafer stages. Alternatively, thesituation can be reversed, with the interferometry system attached to amovable object and the measurement object attached to a stationaryobject.

More generally, such interferometry systems can be used to measure theposition of any one component of the exposure system relative to anyother component of the exposure system, in which the interferometrysystem is attached to, or supported by, one of the components and themeasurement object is attached, or is supported by the other of thecomponents.

Another example of a lithography tool 1100 using an interferometrysystem 1126 is shown in FIG. 5. The interferometry system is used toprecisely measure the position of a wafer (not shown) within an exposuresystem. Here, stage 1122 is used to position and support the waferrelative to an exposure station. Scanner 1100 includes a frame 1102,which carries other support structures and various components carried onthose structures. An exposure base 1104 has mounted on top of it a lenshousing 1106 atop of which is mounted a reticle or mask stage 1116,which is used to support a reticle or mask. A positioning system forpositioning the mask relative to the exposure station is indicatedschematically by element 1117. Positioning system 1117 can include,e.g., piezoelectric transducer elements and corresponding controlelectronics. Although, it is not included in this described embodiment,one or more of the interferometry systems described above can also beused to precisely measure the position of the mask stage as well asother moveable elements whose position must be accurately monitored inprocesses for fabricating lithographic structures (see supra Sheats andSmith Microlithography: Science and Technology).

Suspended below exposure base 1104 is a support base 1113 that carrieswafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflectinga measurement beam 1154 directed to the stage by interferometry system1126. A positioning system for positioning stage 1122 relative tointerferometry system 1126 is indicated schematically by element 1119.Positioning system 1119 can include, e.g., piezoelectric transducerelements and corresponding control electronics. The measurement beamreflects back to the interferometry system, which is mounted on exposurebase 1104. The interferometry system can be any of the embodimentsdescribed previously.

During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beamfrom a UV laser (not shown), passes through a beam shaping opticsassembly 1112 and travels downward after reflecting from mirror 1114.Thereafter, the radiation beam passes through a mask (not shown) carriedby mask stage 1116. The mask (not shown) is imaged onto a wafer (notshown) on wafer stage 1122 via a lens assembly 1108 carried in a lenshousing 1106. Base 1104 and the various components supported by it areisolated from environmental vibrations by a damping system depicted byspring 1120.

In other embodiments of the lithographic scanner, one or more of theinterferometry systems described previously can be used to measuredistance along multiple axes and angles associated for example with, butnot limited to, the wafer and reticle (or mask) stages. Also, ratherthan a UV laser beam, other beams can be used to expose the waferincluding, e.g., x-ray beams, electron beams, ion beams, and visibleoptical beams.

In some embodiments, the lithographic scanner can include what is knownin the art as a column reference. In such embodiments, theinterferometry system 1126 directs the reference beam (not shown) alongan external reference path that contacts a reference mirror (not shown)mounted on some structure that directs the radiation beam, e.g., lenshousing 1106. The reference mirror reflects the reference beam back tothe interferometry system. The interference signal produce byinterferometry system 1126 when combining measurement beam 1154reflected from stage 1122 and the reference beam reflected from areference mirror mounted on the lens housing 1106 indicates changes inthe position of the stage relative to the radiation beam. Furthermore,in other embodiments the interferometry system 1126 can be positioned tomeasure changes in the position of reticle (or mask) stage 1116 or othermovable components of the scanner system. Finally, the interferometrysystems can be used in a similar fashion with lithography systemsinvolving steppers, in addition to, or rather than, scanners.

As is well known in the art, lithography is a critical part ofmanufacturing methods for making semiconducting devices. For example,U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods.These steps are described below with reference to FIGS. 6 a and 6 b.FIG. 6 a is a flow chart of the sequence of manufacturing asemiconductor device such as a semiconductor chip (e.g., IC or LSI), aliquid crystal panel or a CCD. Step 1151 is a design process fordesigning the circuit of a semiconductor device. Step 1152 is a processfor manufacturing a mask on the basis of the circuit pattern design.Step 1153 is a process for manufacturing a wafer by using a materialsuch as silicon.

Step 1154 is a wafer process which is called a pre-process wherein, byusing the so prepared mask and wafer, circuits are formed on the waferthrough lithography. To form circuits on the wafer that correspond withsufficient spatial resolution those patterns on the mask,interferometric positioning of the lithography tool relative the waferis necessary. The interferometry methods and systems described hereincan be especially useful to improve the effectiveness of the lithographyused in the wafer process.

Step 1155 is an assembling step, which is called a post-process whereinthe wafer processed by step 1154 is formed into semiconductor chips.This step includes assembling (dicing and bonding) and packaging (chipsealing). Step 1156 is an inspection step wherein operability check,durability check and so on of the semiconductor devices produced by step1155 are carried out. With these processes, semiconductor devices arefinished and they are shipped (step 1157).

FIG. 6 b is a flow chart showing details of the wafer process. Step 1161is an oxidation process for oxidizing the surface of a wafer. Step 1162is a CVD process for forming an insulating film on the wafer surface.Step 1163 is an electrode forming process for forming electrodes on thewafer by vapor deposition. Step 1164 is an ion implanting process forimplanting ions to the wafer. Step 1165 is a resist process for applyinga resist (photosensitive material) to the wafer. Step 1166 is anexposure process for printing, by exposure (i.e., lithography), thecircuit pattern of the mask on the wafer through the exposure apparatusdescribed above. Once again, as described above, the use of theinterferometry systems and methods described herein improve the accuracyand resolution of such lithography steps.

Step 1167 is a developing process for developing the exposed wafer. Step1168 is an etching process for removing portions other than thedeveloped resist image. Step 1169 is a resist separation process forseparating the resist material remaining on the wafer after beingsubjected to the etching process. By repeating these processes, circuitpatterns are formed and superimposed on the wafer.

The interferometry systems described above can also be used in otherapplications in which the relative position of an object needs to bemeasured precisely. For example, in applications in which a write beamsuch as a laser, x-ray, ion, or electron beam, marks a pattern onto asubstrate as either the substrate or beam moves, the interferometrysystems can be used to measure the relative movement between thesubstrate and write beam.

As an example, a schematic of a beam writing system 1200 is shown inFIG. 7. A source 1210 generates a write beam 1212, and a beam focusingassembly 1214 directs the radiation beam to a substrate 1216 supportedby a movable stage 1218. To determine the relative position of thestage, an interferometry system 1220 directs a reference beam 1222 to amirror 1224 mounted on beam focusing assembly 1214 and a measurementbeam 1226 to a mirror 1228 mounted on stage 1218. Since the referencebeam contacts a mirror mounted on the beam focusing assembly, the beamwriting system is an example of a system that uses a column reference.Interferometry system 1220 can be any of the interferometry systemsdescribed previously. Changes in the position measured by theinterferometry system correspond to changes in the relative position ofwrite beam 1212 on substrate 1216. Interferometry system 1220 sends ameasurement signal 1232 to controller 1230 that is indicative of therelative position of write beam 1212 on substrate 1216. Controller 1230sends an output signal 1234 to a base 1236 that supports and positionsstage 1218. In addition, controller 1230 sends a signal 1238 to source1210 to vary the intensity of, or block, write beam 1212 so that thewrite beam contacts the substrate with an intensity sufficient to causephotophysical or photochemical change only at selected positions of thesubstrate.

Furthermore, in some embodiments, controller 1230 can cause beamfocusing assembly 1214 to scan the write beam over a region of thesubstrate, e.g., using signal 1244. As a result, controller 1230 directsthe other components of the system to pattern the substrate. Thepatterning is typically based on an electronic design pattern stored inthe controller. In some applications the write beam patterns a resistcoated on the substrate and in other applications the write beamdirectly patterns, e.g., etches, the substrate.

An important application of such a system is the fabrication of masksand reticles used in the lithography methods described previously. Forexample, to fabricate a lithography mask an electron beam can be used topattern a chromium-coated glass substrate. In such cases where the writebeam is an electron beam, the beam writing system encloses the electronbeam path in a vacuum. Also, in cases where the write beam is, e.g., anelectron or ion beam, the beam focusing assembly includes electric fieldgenerators such as quadrapole lenses for focusing and directing thecharged particles onto the substrate under vacuum. In other cases wherethe write beam is a radiation beam, e.g., x-ray, UV, or visibleradiation, the beam focusing assembly includes corresponding optics andfor focusing and directing the radiation to the substrate.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.Accordingly, other embodiments are within the scope of the followingclaims.

1. A method, comprising: locating a plurality of alignment marks on amoveable stage; interferometrically measuring a position of ameasurement object along an axis of an interferometer for each of thealignment mark locations by simultaneously directing three measurementbeams to contact the measurement object at a common location and to formthree output beams from the measurement beams, each output beamincluding interferometric information about a distance between theinterferometer and the measurement object along a respective axis; andderiving information about a surface figure of the measurement objectusing the interferometric position measurements, wherein the position ofthe measurement object is measured using an interferometry assembly ofthe interferometer and either the measurement object or theinterferometry assembly are attached to the stage.
 2. The method ofclaim 1 wherein the information comprises information related to acertain spatial frequency component of the surface figure of themeasurement object.
 3. The method of claim 1 further comprisingdetermining the surface figure of the measurement object using theinformation.
 4. The method of claim 3 wherein determining the surfacefigure of the measurement object comprises using values of a parameterassociated with a displacement of the measurement object along threedifferent measurement axes in addition to the information.
 5. The methodof claim 4 wherein the parameter values are determined byinterferometrically monitoring the displacement of the measurementobject along each of the three different interferometer axes whilemoving the measurement object relative to the interferometry assembly;and determining the parameter values for different positions of themeasurement object from the monitored displacements, wherein for a givenposition the parameter is based on the displacements of the measurementobject along each of the three different interferometer axes at thegiven position.
 6. The method of claim 5 wherein determining the surfacefigure of the measurement object comprises frequency transforming theparameter values.
 7. The method of claim 6 wherein the frequencytransform is a Fourier transform.
 8. The method of claim 6 wherein theinformation comprises information related to a certain spatial frequencycomponent of the surface figure of the measurement object at which thefrequency transform of the parameter values is substantiallyinsensitive.
 9. The method of claim 4 wherein the parameter is a seconddifference parameter.
 10. The method of claim 1 wherein the plurality ofalignment marks comprises a linear array of alignment marks.
 11. Themethod of claim 10 wherein the linear array of the alignment marks arenominally parallel to the measurement object during the measuring. 12.The method of claim 10 wherein adjacent alignment marks in the lineararray are separated by a distance, d_(a), and the information about thesurface figure of the measurement object is related to a spatialfrequency component of the surface figure having a spatial wavelength Λgreater than d_(a).
 13. The method of claim 12 wherein Λ=4d_(a).
 14. Themethod of claim 1 wherein the alignment marks are located on the surfaceof an object supported by the moveable stage, and locating the alignmentmarks includes locating the alignment marks for at least two differentpositions of the object.
 15. The method of claim 14 wherein locating thealignment marks comprises rotating the object 180° from a first positionto a second position.
 16. The method of claim 14 wherein locating thealignment marks comprises translating the object from a first positionto a second position.
 17. The method of claim 16 wherein the translationis by an amount related to a spacing of the alignment marks.
 18. Themethod of claim 1 further comprising improving the accuracy ofmeasurements made using the interferometry assembly and the measurementobject by using the information about the surface figure of themeasurement object.
 19. The method of claim 1 further comprisingexposing a substrate supported by the moveable stage with a radiationpattern while interferometrically monitoring a distance between theinterferometry assembly and the measurement object using a lithographytool, wherein the position of the substrate relative to a referenceframe is related to the distance between the interferometry assembly andthe measurement object.
 20. A lithography method for use in fabricatingintegrated circuits on a wafer, the method comprising: supporting thewafer on a moveable stage; imaging spatially patterned radiation ontothe wafer; adjusting the position of the stage; and monitoring theposition of the stage using a measurement object and information aboutthe surface figure of the measurement object derived using the method ofclaim 1 to improve the accuracy of the monitored position of the stage.21. A lithography method for use in the fabrication of integratedcircuits comprising: directing input radiation through a mask to producespatially patterned radiation; positioning the mask relative to theinput radiation; monitoring the position of the mask relative to theinput radiation using a measurement object and information about thesurface figure of the measurement object derived using the method ofclaim 1 to improve the accuracy of the monitored position of the mask;and imaging the spatially patterned radiation onto a wafer.
 22. Alithography method for fabricating integrated circuits on a wafercomprising: positioning a first component of a lithography systemrelative to a second component of a lithography system to expose thewafer to spatially patterned radiation; and monitoring the position ofthe first component relative to the second component using a measurementobject and using information about the surface figure of the measurementobject derived using the method of claim 1 to improve the accuracy ofthe monitored position of the first component.
 23. A method forfabricating integrated circuits comprising: applying a resist to awafer; forming a pattern of a mask in the resist by exposing the waferto radiation using the lithography method of claim 20; and producing anintegrated circuit from the wafer.
 24. A method for fabricatingintegrated circuits comprising: applying a resist to a wafer; forming apattern of a mask in the resist by exposing the wafer to radiation usingthe lithography method of claim 21; and producing an integrated circuitfrom the wafer.
 25. A method for fabricating integrated circuitscomprising: applying a resist to a wafer; forming a pattern of a mask inthe resist by exposing the wafer to radiation using the lithographymethod of claim 22; and producing an integrated circuit from the wafer.26. A system, comprising: a moveable stage; an alignment sensorconfigured to locate alignment marks associated with the moveable stage;a interferometer assembly configured to simultaneously direct threemeasurement beams to contact a measurement object at a common locationand produce three output beams from the measurement beams, each outputbeam including interferometric information about a distance between theinterferometer and a measurement object along a respective axis, theinterferometer assembly or the measurement object being attached to themoveable stage; and an electronic processor configured to deriveinformation about a surface figure of the measurement object based ondata acquired by locating the plurality of alignment marks with thealignment sensor and measuring the position of the measurement objectalong one of the respective axes of the interferometer assembly for eachof the alignment mark locations.
 27. The system of claim 26 wherein themeasurement object is a plane mirror.
 28. The system of claim 26 whereina surface of the stage includes the alignment marks.
 29. The system ofclaim 26 wherein the stage is configured to support an object thatincludes the alignment marks.
 30. The system of claim 26 wherein thealignment sensor is an optical alignment sensor.
 31. The system of claim26 wherein the optical alignment sensor comprises a microscope.
 32. Alithography system for use in fabricating integrated circuits on awafer, the system comprising: the system of claim 26; an illuminationsystem for imaging spatially patterned radiation onto a wafer supportedby the moveable stage; and a positioning system for adjusting theposition of the stage relative to the imaged radiation; wherein theinterferometer assembly is configured to monitor the position of thewafer relative to the imaged radiation and electronic processor isconfigured to use the information about the surface figure of themeasurement object to improve the accuracy of the monitored position ofthe wafer.
 33. A method for fabricating integrated circuits comprising:applying a resist to a wafer; forming a pattern of a mask in the resistby exposing the wafer to radiation with the lithography system of claim32, wherein exposing the wafer comprises positioning the wafer relativeto the illumination system using the positioning system, directingradiation to the wafer using the illumination system, and monitoring theposition of the wafer relative to the radiation using the interferometerassembly; and producing an integrated circuit from the wafer.
 34. A beamwriting system for use in fabricating a lithography mask, the systemcomprising: the system of claim 26; a source providing a write beam topattern a substrate supported by the moveable stage; a beam directingassembly for delivering the write beam to the substrate; a positioningsystem for positioning the stage and beam directing assembly relativeone another; wherein the interferometer assembly is configured tomonitor the position of the stage relative to the beam directingassembly and electronic processor is configured to use the informationabout the surface figure of the measurement object to improve theaccuracy of the monitored position of the stage.
 35. A method forfabricating a lithography mask comprising: directing a beam to asubstrate with the beam writing system of claim 34, wherein directingthe beam comprises positioning the substrate relative to the beamdirecting assembly using the Positioning system, exposing the substrateto the beam using the source and the beam directing assembly, andmonitoring the position of the substrate with respect to the beamdirecting assembly using the interferometer assembly; varying theintensity or the position of the beam at the substrate to form a patternin the substrate; and forming the lithography mask from the patternedsubstrate.
 36. A method, comprising: locating a plurality of alignmentmarks on a moveable stage; interferometrically measuring a position of ameasurement object along an interferometer axis for each of thealignment mark locations; deriving information about a surface figure ofthe measurement object using the interferometric position measurements;and determining the surface figure of the measurement object using theinformation, wherein the position of the measurement object is measuredusing an interferometry assembly and either the measurement object orthe interferometry assembly are attached to the stage and determiningthe surface figure of the measurement object comprises using values of aparameter associated with a displacement of the measurement object alongthree different measurement axes in addition to the information, wherethe parameter values are determined by interferometrically monitoringthe displacement of the measurement object along each of the threedifferent interferometer axes while moving the measurement objectrelative to the interferometry assembly and determining the parametervalues for different positions of the measurement object from themonitored displacements, wherein for a given position the parameter isbased on the displacements of the measurement object along each of thethree different interferometer axes at the given position, and wheredetermining the surface figure of the measurement object comprisesfrequency transforming the parameter values.
 37. The method of claim 36wherein the frequency transform is a Fourier transform.
 38. The methodof claim 36 wherein the information comprises information related to acertain spatial frequency component of the surface figure of themeasurement object at which the frequency transform of the parametervalues is substantially insensitive.
 39. The method of claim 36 whereinthe parameter is a second difference parameter.